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20310:{\displaystyle {\begin{aligned}d\operatorname {tr} (\mathbf {AXBX^{\top }C} )&=d\operatorname {tr} \left(\mathbf {CAXBX^{\top }} \right)=\operatorname {tr} \left(d\left(\mathbf {CAXBX^{\top }} \right)\right)\\&=\operatorname {tr} \left(\mathbf {CAX} d(\mathbf {BX^{\top }} \right)+d\left(\mathbf {CAX} )\mathbf {BX^{\top }} \right)\\&=\operatorname {tr} \left(\mathbf {CAX} d\left(\mathbf {BX^{\top }} \right)\right)+\operatorname {tr} \left(d(\mathbf {CAX} )\mathbf {BX^{\top }} \right)\\&=\operatorname {tr} \left(\mathbf {CAXB} d\left(\mathbf {X^{\top }} \right)\right)+\operatorname {tr} \left(\mathbf {CA} (d\mathbf {X} )\mathbf {BX^{\top }} \right)\\&=\operatorname {tr} \left(\mathbf {CAXB} (d\mathbf {X} )^{\top }\right)+\operatorname {tr} (\mathbf {CA} \left(d\mathbf {X} )\mathbf {BX^{\top }} \right)\\&=\operatorname {tr} \left(\left(\mathbf {CAXB} (d\mathbf {X} )^{\top }\right)^{\top }\right)+\operatorname {tr} \left(\mathbf {CA} (d\mathbf {X} )\mathbf {BX^{\top }} \right)\\&=\operatorname {tr} \left((d\mathbf {X} )\mathbf {B^{\top }X^{\top }A^{\top }C^{\top }} \right)+\operatorname {tr} \left(\mathbf {CA} (d\mathbf {X} )\mathbf {BX^{\top }} \right)\\&=\operatorname {tr} \left(\mathbf {B^{\top }X^{\top }A^{\top }C^{\top }} (d\mathbf {X} )\right)+\operatorname {tr} \left(\mathbf {BX^{\top }} \mathbf {CA} (d\mathbf {X} )\right)\\&=\operatorname {tr} \left(\left(\mathbf {B^{\top }X^{\top }A^{\top }C^{\top }} +\mathbf {BX^{\top }} \mathbf {CA} \right)d\mathbf {X} \right)\\&=\operatorname {tr} \left(\left(\mathbf {CAXB} +\mathbf {A^{\top }C^{\top }XB^{\top }} \right)^{\top }d\mathbf {X} \right)\end{aligned}}}
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7385:
10197:{\displaystyle {\begin{aligned}{\frac {\partial y}{\partial \mathbf {x} }}&={\begin{bmatrix}{\frac {\partial y}{\partial x_{1}}}\\{\frac {\partial y}{\partial x_{2}}}\\\vdots \\{\frac {\partial y}{\partial x_{n}}}\\\end{bmatrix}}.\\{\frac {\partial \mathbf {y} }{\partial x}}&={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x}}&{\frac {\partial y_{2}}{\partial x}}&\cdots &{\frac {\partial y_{m}}{\partial x}}\end{bmatrix}}.\\{\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}&={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x_{1}}}&{\frac {\partial y_{2}}{\partial x_{1}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{1}}}\\{\frac {\partial y_{1}}{\partial x_{2}}}&{\frac {\partial y_{2}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{2}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{1}}{\partial x_{n}}}&{\frac {\partial y_{2}}{\partial x_{n}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{n}}}\\\end{bmatrix}}.\\{\frac {\partial y}{\partial \mathbf {X} }}&={\begin{bmatrix}{\frac {\partial y}{\partial x_{11}}}&{\frac {\partial y}{\partial x_{12}}}&\cdots &{\frac {\partial y}{\partial x_{1q}}}\\{\frac {\partial y}{\partial x_{21}}}&{\frac {\partial y}{\partial x_{22}}}&\cdots &{\frac {\partial y}{\partial x_{2q}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y}{\partial x_{p1}}}&{\frac {\partial y}{\partial x_{p2}}}&\cdots &{\frac {\partial y}{\partial x_{pq}}}\\\end{bmatrix}}.\end{aligned}}}
8470:{\displaystyle {\begin{aligned}{\frac {\partial y}{\partial \mathbf {x} }}&={\begin{bmatrix}{\frac {\partial y}{\partial x_{1}}}&{\frac {\partial y}{\partial x_{2}}}&\cdots &{\frac {\partial y}{\partial x_{n}}}\end{bmatrix}}.\\{\frac {\partial \mathbf {y} }{\partial x}}&={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x}}\\{\frac {\partial y_{2}}{\partial x}}\\\vdots \\{\frac {\partial y_{m}}{\partial x}}\\\end{bmatrix}}.\\{\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}&={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x_{1}}}&{\frac {\partial y_{1}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{1}}{\partial x_{n}}}\\{\frac {\partial y_{2}}{\partial x_{1}}}&{\frac {\partial y_{2}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{2}}{\partial x_{n}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{m}}{\partial x_{1}}}&{\frac {\partial y_{m}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{n}}}\\\end{bmatrix}}.\\{\frac {\partial y}{\partial \mathbf {X} }}&={\begin{bmatrix}{\frac {\partial y}{\partial x_{11}}}&{\frac {\partial y}{\partial x_{21}}}&\cdots &{\frac {\partial y}{\partial x_{p1}}}\\{\frac {\partial y}{\partial x_{12}}}&{\frac {\partial y}{\partial x_{22}}}&\cdots &{\frac {\partial y}{\partial x_{p2}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y}{\partial x_{1q}}}&{\frac {\partial y}{\partial x_{2q}}}&\cdots &{\frac {\partial y}{\partial x_{pq}}}\\\end{bmatrix}}.\end{aligned}}}
9096:
8486:
9091:{\displaystyle {\begin{aligned}{\frac {\partial \mathbf {Y} }{\partial x}}&={\begin{bmatrix}{\frac {\partial y_{11}}{\partial x}}&{\frac {\partial y_{12}}{\partial x}}&\cdots &{\frac {\partial y_{1n}}{\partial x}}\\{\frac {\partial y_{21}}{\partial x}}&{\frac {\partial y_{22}}{\partial x}}&\cdots &{\frac {\partial y_{2n}}{\partial x}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{m1}}{\partial x}}&{\frac {\partial y_{m2}}{\partial x}}&\cdots &{\frac {\partial y_{mn}}{\partial x}}\\\end{bmatrix}}.\\d\mathbf {X} &={\begin{bmatrix}dx_{11}&dx_{12}&\cdots &dx_{1n}\\dx_{21}&dx_{22}&\cdots &dx_{2n}\\\vdots &\vdots &\ddots &\vdots \\dx_{m1}&dx_{m2}&\cdots &dx_{mn}\\\end{bmatrix}}.\end{aligned}}}
3991:
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4183:
3986:{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x_{1}}}&{\frac {\partial y_{1}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{1}}{\partial x_{n}}}\\{\frac {\partial y_{2}}{\partial x_{1}}}&{\frac {\partial y_{2}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{2}}{\partial x_{n}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{m}}{\partial x_{1}}}&{\frac {\partial y_{m}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{n}}}\\\end{bmatrix}}.}
34527:
4962:{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}={\begin{bmatrix}{\frac {\partial y}{\partial x_{11}}}&{\frac {\partial y}{\partial x_{21}}}&\cdots &{\frac {\partial y}{\partial x_{p1}}}\\{\frac {\partial y}{\partial x_{12}}}&{\frac {\partial y}{\partial x_{22}}}&\cdots &{\frac {\partial y}{\partial x_{p2}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y}{\partial x_{1q}}}&{\frac {\partial y}{\partial x_{2q}}}&\cdots &{\frac {\partial y}{\partial x_{pq}}}\\\end{bmatrix}}.}
4549:{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}={\begin{bmatrix}{\frac {\partial y_{11}}{\partial x}}&{\frac {\partial y_{12}}{\partial x}}&\cdots &{\frac {\partial y_{1n}}{\partial x}}\\{\frac {\partial y_{21}}{\partial x}}&{\frac {\partial y_{22}}{\partial x}}&\cdots &{\frac {\partial y_{2n}}{\partial x}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{m1}}{\partial x}}&{\frac {\partial y_{m2}}{\partial x}}&\cdots &{\frac {\partial y_{mn}}{\partial x}}\\\end{bmatrix}}.}
29124:
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1065:). However, even within a given field different authors can be found using competing conventions. Authors of both groups often write as though their specific conventions were standard. Serious mistakes can result when combining results from different authors without carefully verifying that compatible notations have been used. Definitions of these two conventions and comparisons between them are collected in the
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done in this notation without use of the single-variable matrix notation. However, many problems in estimation theory and other areas of applied mathematics would result in too many indices to properly keep track of, pointing in favor of matrix calculus in those areas. Also, Einstein notation can be very useful in proving the identities presented here (see section on
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29119:{\displaystyle \mathbf {U} ^{-1}\left({\frac {\partial \mathbf {U} }{\partial x}}\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial y}}-{\frac {\partial ^{2}\mathbf {U} }{\partial x\partial y}}+{\frac {\partial \mathbf {U} }{\partial y}}\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\right)\mathbf {U} ^{-1}}
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denominator layout rarely occurs. In cases involving matrices where it makes sense, we give numerator-layout and mixed-layout results. As noted above, cases where vector and matrix denominators are written in transpose notation are equivalent to numerator layout with the denominators written without the transpose.
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Keep in mind that various authors use different combinations of numerator and denominator layouts for different types of derivatives, and there is no guarantee that an author will consistently use either numerator or denominator layout for all types. Match up the formulas below with those quoted in
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There are two types of derivatives with matrices that can be organized into a matrix of the same size. These are the derivative of a matrix by a scalar and the derivative of a scalar by a matrix. These can be useful in minimization problems found in many areas of applied mathematics and have adopted
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Here, we have used the term "matrix" in its most general sense, recognizing that vectors are simply matrices with one column (and scalars are simply vectors with one row). Moreover, we have used bold letters to indicate vectors and bold capital letters for matrices. This notation is used throughout.
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Not all math textbooks and papers are consistent in this respect throughout. That is, sometimes different conventions are used in different contexts within the same book or paper. For example, some choose denominator layout for gradients (laying them out as column vectors), but numerator layout for
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Matrix calculus refers to a number of different notations that use matrices and vectors to collect the derivative of each component of the dependent variable with respect to each component of the independent variable. In general, the independent variable can be a scalar, a vector, or a matrix while
2252:
convention is very similar to the matrix calculus, except one writes only a single component at a time. It has the advantage that one can easily manipulate arbitrarily high rank tensors, whereas tensors of rank higher than two are quite unwieldy with matrix notation. All of the work here can be
2230:
Despite the use of the terms "numerator layout" and "denominator layout", there are actually more than two possible notational choices involved. The reason is that the choice of numerator vs. denominator (or in some situations, numerator vs. mixed) can be made independently for scalar-by-vector,
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Each of the previous two cases can be considered as an application of the derivative of a vector with respect to a vector, using a vector of size one appropriately. Similarly we will find that the derivatives involving matrices will reduce to derivatives involving vectors in a corresponding way.
2234:
The choice of numerator layout in the introductory sections below does not imply that this is the "correct" or "superior" choice. There are advantages and disadvantages to the various layout types. Serious mistakes can result from carelessly combining formulas written in different layouts, and
1053:. Both of these conventions are possible even when the common assumption is made that vectors should be treated as column vectors when combined with matrices (rather than row vectors). A single convention can be somewhat standard throughout a single field that commonly uses matrix calculus (e.g.
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There are a total of nine possibilities using scalars, vectors, and matrices. Notice that as we consider higher numbers of components in each of the independent and dependent variables we can be left with a very large number of possibilities. The six kinds of derivatives that can be most neatly
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When taking derivatives with an aggregate (vector or matrix) denominator in order to find a maximum or minimum of the aggregate, it should be kept in mind that using numerator layout will produce results that are transposed with respect to the aggregate. For example, in attempting to find the
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This section discusses the similarities and differences between notational conventions that are used in the various fields that take advantage of matrix calculus. Although there are largely two consistent conventions, some authors find it convenient to mix the two conventions in forms that are
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is defined in terms of a scalar variable, and then a derivative of a scalar function of the curve is taken with respect to the scalar that parameterizes the curve.) For each of the various combinations, we give numerator-layout and denominator-layout results, except in the cases above where
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18988:{\displaystyle {\begin{aligned}\operatorname {tr} (\mathbf {A} )&=\operatorname {tr} \left(\mathbf {A^{\top }} \right)\\\operatorname {tr} (\mathbf {ABCD} )&=\operatorname {tr} (\mathbf {BCDA} )=\operatorname {tr} (\mathbf {CDAB} )=\operatorname {tr} (\mathbf {DABC} )\end{aligned}}}
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converting from one layout to another requires care to avoid errors. As a result, when working with existing formulas the best policy is probably to identify whichever layout is used and maintain consistency with it, rather than attempting to use the same layout in all situations.
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27514:{\displaystyle {\begin{aligned}\left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|{\Big (}&\left(\mathbf {X^{\top }AX} \right)^{-1}\mathbf {X^{\top }A} +{}\\&\left(\mathbf {X^{\top }A^{\top }X} \right)^{-1}\mathbf {X^{\top }A^{\top }} {\Big )}\end{aligned}}}
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10227:. The sum rule applies universally, and the product rule applies in most of the cases below, provided that the order of matrix products is maintained, since matrix products are not commutative. The chain rule applies in some of the cases, but unfortunately does
1638:{\displaystyle \nabla f=\left({\frac {\partial f}{\partial \mathbf {x} }}\right)^{\mathsf {T}}={\begin{bmatrix}{\dfrac {\partial f}{\partial x_{1}}}&{\dfrac {\partial f}{\partial x_{2}}}&{\dfrac {\partial f}{\partial x_{3}}}\\\end{bmatrix}}^{\textsf {T}}.}
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This is presented first because all of the operations that apply to vector-by-vector differentiation apply directly to vector-by-scalar or scalar-by-vector differentiation simply by reducing the appropriate vector in the numerator or denominator to a scalar.
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27700:{\displaystyle {\begin{aligned}\left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|{\Big (}&\mathbf {AX} \left(\mathbf {X^{\top }AX} \right)^{-1}+{}\\&\mathbf {A^{\top }X} \left(\mathbf {X^{\top }A^{\top }X} \right)^{-1}{\Big )}\end{aligned}}}
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1652:, which collects the derivative with respect to each matrix element in the corresponding position in the resulting matrix. In that case the scalar must be a function of each of the independent variables in the matrix. As another example, if we have an
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33354:{\displaystyle \sum _{ij}\mathbf {P} _{i}(d\mathbf {X} )\mathbf {P} _{j}{\begin{cases}f'(\lambda _{i})&\lambda _{i}=\lambda _{j}\\{\frac {f(\lambda _{i})-f(\lambda _{j})}{\lambda _{i}-\lambda _{j}}}&\lambda _{i}\neq \lambda _{j}\end{cases}}}
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21210:
15534:{\displaystyle {\frac {\partial (\mathbf {x} \cdot \mathbf {x} )}{\partial \mathbf {x} }}={\frac {\partial \mathbf {x} ^{\top }\mathbf {x} }{\partial \mathbf {x} }}={\frac {\partial \left\Vert \mathbf {x} \right\Vert ^{2}}{\partial \mathbf {x} }}=}
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do not exist when applied to matrix-valued functions of matrices. However, the product rule of this sort does apply to the differential form (see below), and this is the way to derive many of the identities below involving the
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18775:(whose outputs are matrices) assume the matrices are laid out consistent with the vector layout, i.e. numerator-layout matrix when numerator-layout vector and vice versa; otherwise, transpose the vector-by-vector derivatives.
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The three types of derivatives that have not been considered are those involving vectors-by-matrices, matrices-by-vectors, and matrices-by-matrices. These are not as widely considered and a notation is not widely agreed upon.
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2983:{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}={\begin{bmatrix}{\dfrac {\partial y}{\partial x_{1}}}&{\dfrac {\partial y}{\partial x_{2}}}&\cdots &{\dfrac {\partial y}{\partial x_{n}}}\end{bmatrix}}.}
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Giles, Mike B. (2008). "Collected matrix derivative results for forward and reverse mode algorithmic differentiation". In
Bischof, Christian H.; Bücker, H. Martin; Hovland, Paul; Naumann, Uwe; Utke, Jean (eds.).
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1988:
to take derivatives with respect to vectors. In the case that a matrix function of a matrix is Fréchet differentiable, the two derivatives will agree up to translation of notations. As is the case in general for
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5670:) and follow the numerator layout. This makes it possible to claim that the matrix is laid out according to both numerator and denominator. In practice this produces results the same as the numerator layout.
3166:{\displaystyle \nabla f={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}\\\vdots \\{\frac {\partial f}{\partial x_{n}}}\end{bmatrix}}=\left({\frac {\partial f}{\partial \mathbf {x} }}\right)^{\mathsf {T}}}
29425:
28746:
23785:{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {AX^{\top }} \right)}{\partial \mathbf {X} }}={\frac {\partial \operatorname {tr} \left(\mathbf {X^{\top }A} \right)}{\partial \mathbf {X} }}=}
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20995:
12754:
33816:
30603:
29598:
26193:
24186:
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15291:
15144:
13039:
12325:{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}{\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}{\frac {\partial \mathbf {f} (\mathbf {g} )}{\partial \mathbf {g} }}}
12213:{\displaystyle {\frac {\partial \mathbf {f} (\mathbf {g} )}{\partial \mathbf {g} }}{\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}
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12623:
1967:
of rank higher than 2, so that they do not fit neatly into a matrix. In the following three sections we will define each one of these derivatives and relate them to other branches of mathematics. See the
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6234:
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5756:
5559:
27759:
20653:
16864:
15941:
7245:
7047:
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6905:
6708:
6664:{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }},{\frac {\partial \mathbf {y} }{\partial x}},{\frac {\partial \mathbf {y} }{\partial \mathbf {x} }},{\frac {\partial y}{\partial \mathbf {X} }}}
6493:
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6397:
6352:
6192:
6013:
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5885:
5846:
5714:
2674:
1948:
1859:
1814:
1774:
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1963:
Notice that we could also talk about the derivative of a vector with respect to a matrix, or any of the other unfilled cells in our table. However, these derivatives are most naturally organized in a
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18003:
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13987:{\displaystyle \mathbf {u} ^{\top }\mathbf {A} {\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}+\mathbf {v} ^{\top }\mathbf {A} ^{\top }{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}
11299:
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operator applied to matrices). In the latter case, the product rule can't quite be applied directly, either, but the equivalent can be done with a bit more work using the differential identities.
31242:
25331:
140:
34039:
31960:
18375:{\displaystyle {\frac {\partial \mathbf {u} }{\partial x}}{\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}{\frac {\partial \mathbf {f} (\mathbf {g} )}{\partial \mathbf {g} }}}
18265:{\displaystyle {\frac {\partial \mathbf {f} (\mathbf {g} )}{\partial \mathbf {g} }}{\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}{\frac {\partial \mathbf {u} }{\partial x}}}
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22168:
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32497:
13877:{\displaystyle {\frac {\partial (\mathbf {u} \cdot \mathbf {A} \mathbf {v} )}{\partial \mathbf {x} }}={\frac {\partial \mathbf {u} ^{\top }\mathbf {A} \mathbf {v} }{\partial \mathbf {x} }}=}
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2208:
unless otherwise noted. Generally letters from the first half of the alphabet (a, b, c, ...) will be used to denote constants, and from the second half (t, x, y, ...) to denote variables.
31849:
25488:
20461:{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {AXBX^{\top }C} \right)}{\partial \mathbf {X} }}=\mathbf {B^{\top }X^{\top }A^{\top }C^{\top }} +\mathbf {BX^{\top }CA} .}
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6719:
the source to determine the layout used for that particular type of derivative, but be careful not to assume that derivatives of other types necessarily follow the same kind of layout.
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14153:{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}\mathbf {A} \mathbf {v} +{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}\mathbf {A} ^{\top }\mathbf {u} }
4993:
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14595:
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12523:
10888:
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2257:) as an alternative to typical element notation, which can become cumbersome when the explicit sums are carried around. Note that a matrix can be considered a tensor of rank two.
1153:
30141:
26280:
24847:
1324:{\displaystyle \nabla f={\frac {\partial f}{\partial x_{1}}}{\hat {x}}_{1}+{\frac {\partial f}{\partial x_{2}}}{\hat {x}}_{2}+{\frac {\partial f}{\partial x_{3}}}{\hat {x}}_{3},}
1022:
and matrices that can be treated as single entities. This greatly simplifies operations such as finding the maximum or minimum of a multivariate function and solving systems of
33990:
31327:
25440:
25196:
24748:
16600:{\displaystyle {\textbf {D}}^{\top }{\textbf {C}}^{\top }({\textbf {A}}{\textbf {x}}+{\textbf {b}})+{\textbf {A}}^{\top }{\textbf {C}}({\textbf {D}}{\textbf {x}}+{\textbf {e}})}
16470:{\displaystyle ({\textbf {D}}{\textbf {x}}+{\textbf {e}})^{\top }{\textbf {C}}^{\top }{\textbf {A}}+({\textbf {A}}{\textbf {x}}+{\textbf {b}})^{\top }{\textbf {C}}{\textbf {D}}}
31691:
26380:
25814:
1045:
Two competing notational conventions split the field of matrix calculus into two separate groups. The two groups can be distinguished by whether they write the derivative of a
31721:
31619:
27253:
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21145:
12904:
33652:
21815:{\displaystyle \operatorname {tr} \left(\left({\frac {\partial g(\mathbf {U} )}{\partial \mathbf {U} }}\right)^{\top }{\frac {\partial \mathbf {U} }{\partial X_{ij}}}\right)}
19001:
28112:
26547:
21917:
21001:
17488:
14998:
14798:
14651:
12760:
1368:
25088:
24434:
24403:{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {AXB} )}{\partial \mathbf {X} }}={\frac {\partial \operatorname {tr} (\mathbf {BAX} )}{\partial \mathbf {X} }}=}
15598:
15395:
1427:
34651:
34599:
33396:
33076:
26812:{\displaystyle 2\left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|\mathbf {X} ^{-1}=2\left|\mathbf {X^{\top }} \right||\mathbf {A} ||\mathbf {X} |\mathbf {X} ^{-1}}
23843:
23624:
22891:
20788:
20595:{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {AXBX^{\top }C} \right)}{\partial \mathbf {X} }}=\mathbf {CAXB} +\mathbf {A^{\top }C^{\top }XB^{\top }} .}
16975:
14618:
14350:
12546:
10911:
10756:
10668:
10596:
5430:
1453:
23596:{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {AX} )}{\partial \mathbf {X} }}={\frac {\partial \operatorname {tr} (\mathbf {XA} )}{\partial \mathbf {X} }}=}
18654:{\displaystyle \mathbf {v} ^{\top }\times \left({\frac {\partial \mathbf {U} }{\partial x}}\right)+{\frac {\partial \mathbf {v} }{\partial x}}\times \mathbf {U} ^{\top }}
17845:{\displaystyle {\frac {\partial \mathbf {u} }{\partial x}}\times \mathbf {v} +\mathbf {u} ^{\top }\times \left({\frac {\partial \mathbf {v} }{\partial x}}\right)^{\top }}
17742:{\displaystyle \left({\frac {\partial \mathbf {u} }{\partial x}}\right)^{\top }\times \mathbf {v} +\mathbf {u} ^{\top }\times {\frac {\partial \mathbf {v} }{\partial x}}}
30791:{\displaystyle \operatorname {tr} \left(\left({\frac {\partial g(\mathbf {U} )}{\partial \mathbf {U} }}\right)^{\top }{\frac {\partial \mathbf {U} }{\partial x}}\right)}
16340:{\displaystyle {\frac {\partial \;({\textbf {A}}{\textbf {x}}+{\textbf {b}})^{\top }{\textbf {C}}({\textbf {D}}{\textbf {x}}+{\textbf {e}})}{\partial \;{\textbf {x}}}}=}
12045:
25342:
24066:
33946:
31045:
29284:
25627:
23077:{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {U} )}{\partial \mathbf {X} }}+{\frac {\partial \operatorname {tr} (\mathbf {V} )}{\partial \mathbf {X} }}}
30898:
15722:{\displaystyle {\frac {\partial (\mathbf {a} \cdot \mathbf {u} )}{\partial \mathbf {x} }}={\frac {\partial \mathbf {a} ^{\top }\mathbf {u} }{\partial \mathbf {x} }}=}
14560:{\displaystyle {\frac {\partial \mathbf {a} ^{\top }\mathbf {x} }{\partial \mathbf {x} }}={\frac {\partial \mathbf {x} ^{\top }\mathbf {a} }{\partial \mathbf {x} }}=}
14464:{\displaystyle {\frac {\partial (\mathbf {a} \cdot \mathbf {x} )}{\partial \mathbf {x} }}={\frac {\partial (\mathbf {x} \cdot \mathbf {a} )}{\partial \mathbf {x} }}=}
13411:{\displaystyle {\frac {\partial (\mathbf {u} \cdot \mathbf {v} )}{\partial \mathbf {x} }}={\frac {\partial \mathbf {u} ^{\top }\mathbf {v} }{\partial \mathbf {x} }}=}
6710:
separately. We also handle cases of scalar-by-scalar derivatives that involve an intermediate vector or matrix. (This can arise, for example, if a multi-dimensional
4149:
discusses this issue in greater detail. The identities given further down are presented in forms that can be used in conjunction with all common layout conventions.
3300:
3224:
2329:
discusses this issue in greater detail. The identities given further down are presented in forms that can be used in conjunction with all common layout conventions.
23328:
13504:{\displaystyle \mathbf {u} ^{\top }{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}+\mathbf {v} ^{\top }{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}
2039:, using a single variable to represent a large number of variables. In what follows we will distinguish scalars, vectors and matrices by their typeface. We will let
28752:
21286:
13045:
1395:
33684:
23119:
14238:
23179:
22817:
16687:
16626:
31434:{\displaystyle |\mathbf {X} |\operatorname {tr} \left(\mathbf {X} ^{-1}d\mathbf {X} \right)=\operatorname {tr} (\operatorname {adj} (\mathbf {X} )d\mathbf {X} )}
10807:
31640:
15733:
11584:
35545:
Liu, Shuangzhe; Trenkler, Götz; Kollo, Tõnu; von Rosen, Dietrich; Baksalary, Oskar Maria (2023). "Professor Heinz
Neudecker and matrix differential calculus".
31174:
It is often easier to work in differential form and then convert back to normal derivatives. This only works well using the numerator layout. In these rules,
17586:
34866:{\displaystyle {\frac {d\ln au}{dx}}={\frac {1}{au}}{\frac {d(au)}{dx}}={\frac {1}{au}}a{\frac {du}{dx}}={\frac {1}{u}}{\frac {du}{dx}}={\frac {d\ln u}{dx}}.}
30967:
26083:
22720:{\displaystyle \left(\mathbf {C} \mathbf {X} \mathbf {b} \mathbf {a} ^{\top }+\mathbf {C} ^{\top }\mathbf {X} \mathbf {a} \mathbf {b} ^{\top }\right)^{\top }}
18101:
31966:
24585:
22325:{\displaystyle {\frac {\partial (\mathbf {X} \mathbf {a} +\mathbf {b} )^{\top }\mathbf {C} (\mathbf {X} \mathbf {a} +\mathbf {b} )}{\partial \mathbf {X} }}=}
32646:
11671:
36709:
35741:
11817:
6308:, is rarely seen because it makes for ugly formulas that do not correspond to the scalar formulas. As a result, the following layouts can often be found:
1660:
independent variables we might consider the derivative of the dependent vector with respect to the independent vector. The result could be collected in an
32503:
26199:
25554:
18726:
18674:
33405:
32083:
27972:
21701:{\displaystyle \operatorname {tr} \left({\frac {\partial g(\mathbf {U} )}{\partial \mathbf {U} }}{\frac {\partial \mathbf {U} }{\partial X_{ij}}}\right)}
12353:
11487:
10689:
33525:
28372:
23937:
16758:
15830:
14860:
11537:
3362:
This type of notation will be nice when proving product rules and chain rules that come out looking similar to what we are familiar with for the scalar
35727:
27115:{\displaystyle 2\left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|\left(\mathbf {X^{\top }A^{\top }X} \right)^{-1}\mathbf {X^{\top }A^{\top }} }
25002:
18424:
35185:
29713:
28543:
17343:
2599:{\displaystyle {\frac {d\mathbf {y} }{dx}}={\begin{bmatrix}{\frac {dy_{1}}{dx}}\\{\frac {dy_{2}}{dx}}\\\vdots \\{\frac {dy_{m}}{dx}}\\\end{bmatrix}}.}
36697:
28847:
17223:
13653:{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}\mathbf {v} +{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}\mathbf {u} }
11166:
31498:
30687:{\displaystyle \operatorname {tr} \left({\frac {\partial g(\mathbf {U} )}{\partial \mathbf {U} }}{\frac {\partial \mathbf {U} }{\partial x}}\right)}
25723:
33822:
32975:
32286:
31112:
29226:
10212:
As noted above, in general, the results of operations will be transposed when switching between numerator-layout and denominator-layout notation.
29431:
28055:
25890:
17431:
2231:
vector-by-scalar, vector-by-vector, and scalar-by-matrix derivatives, and a number of authors mix and match their layout choices in various ways.
28210:
23997:
17874:
14920:
10948:
28499:{\displaystyle \mathbf {U} \otimes {\frac {\partial \mathbf {V} }{\partial x}}+{\frac {\partial \mathbf {U} }{\partial x}}\otimes \mathbf {V} }
25499:
33519:
22426:{\displaystyle \left(\left(\mathbf {C} +\mathbf {C} ^{\top }\right)(\mathbf {X} \mathbf {a} +\mathbf {b} )\mathbf {a} ^{\top }\right)^{\top }}
17130:
11251:
11073:
5618:
31855:
24669:
18551:{\displaystyle {\frac {\partial \mathbf {U} }{\partial x}}\times \mathbf {v} +\mathbf {U} \times {\frac {\partial \mathbf {v} }{\partial x}}}
17178:
11121:
10997:
10606:
10534:
6112:
5893:
5765:
30230:
26005:
15876:
15786:
10441:
7307:
7263:
7116:
7062:
5568:
5435:
5372:
5226:
1865:
36819:
36704:
34508:
29840:{\displaystyle \mathbf {u} \cdot {\frac {\partial \mathbf {v} }{\partial x}}+{\frac {\partial \mathbf {u} }{\partial x}}\cdot \mathbf {v} }
28670:{\displaystyle \mathbf {U} \circ {\frac {\partial \mathbf {V} }{\partial x}}+{\frac {\partial \mathbf {U} }{\partial x}}\circ \mathbf {V} }
33689:
21551:
17293:
12016:{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}{\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}}
11939:{\displaystyle {\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}
236:
34339:
34102:
22802:{\displaystyle \mathbf {C} \mathbf {X} \mathbf {b} \mathbf {a} ^{\top }+\mathbf {C} ^{\top }\mathbf {X} \mathbf {a} \mathbf {b} ^{\top }}
32913:
30082:
21373:
15364:{\displaystyle {\frac {\partial ^{2}\mathbf {x} ^{\top }\mathbf {A} \mathbf {x} }{\partial \mathbf {x} \partial \mathbf {x} ^{\top }}}=}
15217:{\displaystyle {\frac {\partial ^{2}\mathbf {x} ^{\top }\mathbf {A} \mathbf {x} }{\partial \mathbf {x} \partial \mathbf {x} ^{\top }}}=}
13132:
2019:
36687:
36682:
29378:
28699:
21833:
11445:{\displaystyle v{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}+{\frac {\partial v}{\partial \mathbf {x} }}\mathbf {u} ^{\top }}
1082:, using the broader sense of the term. Matrix notation serves as a convenient way to collect the many derivatives in an organized way.
34045:
23433:
20946:
12705:
3297:
Using the notation just defined for the derivative of a scalar with respect to a vector we can re-write the directional derivative as
36692:
36677:
35791:
30557:
29552:
26143:
24136:
21234:
12993:
33764:
29925:
27923:
16016:{\displaystyle {\frac {\partial \;{\textbf {a}}^{\top }{\textbf {x}}{\textbf {x}}^{\top }{\textbf {b}}}{\partial \;{\textbf {x}}}}=}
35979:
33655:
27799:
22621:{\displaystyle {\frac {\partial (\mathbf {X} \mathbf {a} )^{\top }\mathbf {C} (\mathbf {X} \mathbf {b} )}{\partial \mathbf {X} }}=}
21099:
20824:
17012:
12858:
12583:
6507:
35264:
35216:
35049:{\displaystyle {\frac {d\ln au}{dx}}={\frac {d(\ln a+\ln u)}{dx}}={\frac {d\ln a}{dx}}+{\frac {d\ln u}{dx}}={\frac {d\ln u}{dx}}.}
33575:
27845:
20870:
20698:
17058:
16915:
12629:
12456:
6263:
6197:
6065:
6021:
5719:
5522:
1993:, some formulae may extend under weaker analytic conditions than the existence of the derivative as approximating linear mapping.
1078:
the dependent variable can be any of these as well. Each different situation will lead to a different set of rules, or a separate
36672:
27725:
21521:{\displaystyle {\frac {\partial f(g)}{\partial g}}{\frac {\partial g(u)}{\partial u}}{\frac {\partial u}{\partial \mathbf {X} }}}
20619:
16830:
13280:{\displaystyle {\frac {\partial f(g)}{\partial g}}{\frac {\partial g(u)}{\partial u}}{\frac {\partial u}{\partial \mathbf {x} }}}
7211:
7013:
6920:
6871:
6739:
row vector. Thus, either the results should be transposed at the end or the denominator layout (or mixed layout) should be used.
6674:
6459:
6414:
6363:
6318:
6158:
5979:
5940:
5851:
5812:
5680:
2640:
1914:
1825:
1780:
1740:
32364:
5180:{\displaystyle \nabla _{\mathbf {Y} }f=\operatorname {tr} \left({\frac {\partial f}{\partial \mathbf {X} }}\mathbf {Y} \right).}
34296:
33892:
To convert to normal derivative form, first convert it to one of the following canonical forms, and then use these identities:
28328:{\displaystyle \mathbf {U} {\frac {\partial \mathbf {V} }{\partial x}}+{\frac {\partial \mathbf {U} }{\partial x}}\mathbf {V} }
27212:{\displaystyle 2\left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|\mathbf {AX} \left(\mathbf {X^{\top }AX} \right)^{-1}}
18067:{\displaystyle {\frac {\partial \mathbf {u} }{\partial x}}{\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}}
17992:{\displaystyle {\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}{\frac {\partial \mathbf {u} }{\partial x}}}
11363:{\displaystyle v{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}+\mathbf {u} {\frac {\partial v}{\partial \mathbf {x} }}}
34432:
34246:
34155:
24853:
36289:
36043:
35698:
35617:
35590:
35471:
35369:
14221:{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }},{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}}
14055:{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }},{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}}
13721:{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }},{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}}
13572:{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }},{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}}
11788:{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}+{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}}
32900:{\displaystyle \int _{0}^{\infty }(\mathbf {X} +z\,\mathbf {I} )^{-1}(d\mathbf {X} )(\mathbf {X} +z\,\mathbf {I} )^{-1}\,dz}
26386:
25820:
31201:
25292:
497:
477:
33998:
31923:
30053:{\displaystyle |\mathbf {U} |\operatorname {tr} \left(\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\right)}
29669:{\displaystyle {\frac {\partial g(\mathbf {u} )}{\partial \mathbf {u} }}\cdot {\frac {\partial \mathbf {u} }{\partial x}}}
32152:
22510:{\displaystyle \left(\mathbf {C} +\mathbf {C} ^{\top }\right)(\mathbf {X} \mathbf {a} +\mathbf {b} )\mathbf {a} ^{\top }}
32040:
31248:
35841:
25951:
24991:{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {A} \mathbf {X} ^{n}\right)}{\partial \mathbf {X} }}=}
16103:{\displaystyle {\textbf {x}}^{\top }\left({\textbf {a}}{\textbf {b}}^{\top }+{\textbf {b}}{\textbf {a}}^{\top }\right)}
15228:
15060:
2222:
simply for the purposes of convenience, to avoid overly complicating the discussion. The section after them discusses
973:
536:
32239:
22174:
22139:
22014:
21979:
14748:
14713:
7371:
The results of operations will be transposed when switching between numerator-layout and denominator-layout notation.
6830:
1702:
59:
36787:
36646:
35496:
35446:
35109:
32602:
32459:
29496:
492:
215:
17:
34391:
34205:
33083:
32410:
30297:
26997:{\displaystyle {\frac {\partial \left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|}{\partial \mathbf {X} }}=}
26673:{\displaystyle {\frac {\partial \left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|}{\partial \mathbf {X} }}=}
1980:
The matrix derivative is a convenient notation for keeping track of partial derivatives for doing calculations. The
36201:
36117:
34493:
31731:
24574:{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {AXBX^{\top }C} \right)}{\partial \mathbf {X} }}=}
24440:
6728:
1019:
482:
35645:. Note that this Knowledge article has been nearly completely revised from the version criticized in this article.
34488:
Matrix differential calculus is used in statistics and econometrics, particularly for the statistical analysis of
31815:
30824:
is any polynomial with scalar coefficients, or any matrix function defined by an infinite polynomial series (e.g.
29152:
is any polynomial with scalar coefficients, or any matrix function defined by an infinite polynomial series (e.g.
25446:
36855:
36782:
36714:
36339:
36194:
36162:
35921:
35718:
35138:
31774:
24274:{\displaystyle -\left(\mathbf {X} ^{-1}\right)^{\top }\mathbf {A} ^{\top }\left(\mathbf {X} ^{-1}\right)^{\top }}
23926:{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {X^{\top }AX} \right)}{\partial \mathbf {X} }}=}
23394:
2218:
in vectors and matrices, and no standard appears to be emerging yet. The next two introductory sections use the
813:
487:
467:
149:
26895:{\displaystyle 2\left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|\left(\mathbf {X} ^{-1}\right)^{\top }}
26513:{\displaystyle {\frac {\partial \ln \left|\mathbf {X} ^{\top }\mathbf {X} \right|}{\partial \mathbf {X} ^{+}}}=}
10231:
apply in matrix-by-scalar derivatives or scalar-by-matrix derivatives (in the latter case, mostly involving the
4112:{\displaystyle d\mathbf {f} (\mathbf {v} )={\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}d\mathbf {v} .}
36415:
36392:
36107:
35417:
32968:
22128:{\displaystyle {\frac {\partial \mathbf {a} ^{\top }\mathbf {X} ^{\top }\mathbf {b} }{\partial \mathbf {X} }}=}
16183:{\displaystyle \left({\textbf {a}}{\textbf {b}}^{\top }+{\textbf {b}}{\textbf {a}}^{\top }\right){\textbf {x}}}
15098:
5062:{\displaystyle \nabla _{\mathbf {X} }y(\mathbf {X} )={\frac {\partial y(\mathbf {X} )}{\partial \mathbf {X} }}}
3553:{\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}&x_{2}&\cdots &x_{n}\end{bmatrix}}^{\mathsf {T}}}
3464:{\displaystyle \mathbf {y} ={\begin{bmatrix}y_{1}&y_{2}&\cdots &y_{m}\end{bmatrix}}^{\mathsf {T}}}
2819:{\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}&x_{2}&\cdots &x_{n}\end{bmatrix}}^{\mathsf {T}}}
2426:{\displaystyle \mathbf {y} ={\begin{bmatrix}y_{1}&y_{2}&\cdots &y_{m}\end{bmatrix}}^{\mathsf {T}}}
36505:
36443:
36238:
36112:
35784:
32746:
32196:
1648:
More complicated examples include the derivative of a scalar function with respect to a matrix, known as the
595:
542:
428:
35724:
31447:
30201:{\displaystyle \operatorname {tr} \left(\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\right)}
26553:
26342:{\displaystyle {\frac {\partial \ln \left|\mathbf {X} ^{\top }\mathbf {X} \right|}{\partial \mathbf {X} }}=}
35991:
35969:
35392:
34540:
25251:{\displaystyle {\frac {\partial \operatorname {tr} \left(e^{\mathbf {X} }\right)}{\partial \mathbf {X} }}=}
24803:{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {X} ^{n}\right)}{\partial \mathbf {X} }}=}
15545:
2271:
Because vectors are matrices with only one column, the simplest matrix derivatives are vector derivatives.
254:
226:
36814:
35356:. Lecture Notes in Computational Science and Engineering. Vol. 64. Berlin: Springer. pp. 35–44.
34515:. It is also used in random matrices, statistical moments, local sensitivity and statistical diagnostics.
23796:
23630:
20741:
18796:
function, combined with the fact that the trace function allows transposing and cyclic permutation, i.e.:
14571:
14303:
12499:
10864:
10762:
1096:
337:
36799:
36565:
36179:
36001:
32032:
27314:{\displaystyle {\frac {\partial |\mathbf {X^{\top }} \mathbf {A} \mathbf {X} |}{\partial \mathbf {X} }}=}
24814:
22984:{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {U} +\mathbf {V} )}{\partial \mathbf {X} }}=}
21205:{\displaystyle u{\frac {\partial v}{\partial \mathbf {X} }}+v{\frac {\partial u}{\partial \mathbf {X} }}}
12964:{\displaystyle u{\frac {\partial v}{\partial \mathbf {x} }}+v{\frac {\partial u}{\partial \mathbf {x} }}}
10224:
846:
459:
297:
269:
33952:
31288:
25411:
19066:{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {AXBX^{\top }C} )}{\partial \mathbf {X} }}:}
36184:
35954:
31659:
28166:{\displaystyle {\frac {\partial \mathbf {U} }{\partial x}}+{\frac {\partial \mathbf {V} }{\partial x}}}
26353:
25787:
21968:{\displaystyle {\frac {\partial \mathbf {a} ^{\top }\mathbf {X} \mathbf {b} }{\partial \mathbf {X} }}=}
21055:{\displaystyle {\frac {\partial u}{\partial \mathbf {X} }}+{\frac {\partial v}{\partial \mathbf {X} }}}
17542:{\displaystyle {\frac {\partial \mathbf {u} }{\partial x}}+{\frac {\partial \mathbf {v} }{\partial x}}}
15049:{\displaystyle {\frac {\partial \mathbf {x} ^{\top }\mathbf {A} \mathbf {x} }{\partial \mathbf {x} }}=}
14849:{\displaystyle {\frac {\partial \mathbf {x} ^{\top }\mathbf {A} \mathbf {x} }{\partial \mathbf {x} }}=}
14702:{\displaystyle {\frac {\partial \mathbf {b} ^{\top }\mathbf {A} \mathbf {x} }{\partial \mathbf {x} }}=}
12814:{\displaystyle {\frac {\partial u}{\partial \mathbf {x} }}+{\frac {\partial v}{\partial \mathbf {x} }}}
4014:
717:
681:
463:
342:
231:
221:
35685:
31697:
31590:
25262:
25180:{\displaystyle \sum _{i=0}^{n-1}\left(\mathbf {X} ^{i}\mathbf {A} \mathbf {X} ^{n-i-1}\right)^{\top }}
36603:
36550:
35514:"Matrix differential calculus with applications in the multivariate linear model and its diagnostics"
35086:
34489:
33626:
36011:
33191:
26524:
1337:
322:
36719:
36490:
36038:
35777:
24414:
15578:
15375:
12101:{\displaystyle {\frac {\partial \mathbf {f} (\mathbf {g} (\mathbf {u} ))}{\partial \mathbf {x} }}=}
1400:
616:
181:
35681:
34634:
34582:
33371:
33054:
25400:{\displaystyle {\frac {\partial \operatorname {tr} (\sin(\mathbf {X} ))}{\partial \mathbf {X} }}=}
24125:{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {X^{-1}A} )}{\partial \mathbf {X} }}=}
23826:
23607:
22874:
20771:
16958:
14601:
14333:
12529:
10894:
10739:
10651:
10579:
5413:
5220:
discussed below. After this section, equations will be listed in both competing forms separately.
1436:
36850:
36485:
36157:
35757:
35735:
35702:
34497:
33048:
31101:{\displaystyle {\frac {\partial \operatorname {tr} \left(e^{x\mathbf {A} }\right)}{\partial x}}=}
29352:{\displaystyle \mathbf {A} \mathbf {g} '(x\mathbf {A} )=\mathbf {g} '(x\mathbf {A} )\mathbf {A} }
25697:{\displaystyle \operatorname {cofactor} (X)=|\mathbf {X} |\left(\mathbf {X} ^{-1}\right)^{\top }}
2200:
930:
722:
611:
30956:{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {g} (x\mathbf {A} ))}{\partial x}}=}
23250:
with scalar coefficients, or any matrix function defined by an infinite polynomial series (e.g.
3355:{\displaystyle \nabla _{\mathbf {u} }f={\frac {\partial f}{\partial \mathbf {x} }}\mathbf {u} .}
3287:{\displaystyle \nabla _{\mathbf {u} }{f}(\mathbf {x} )=\nabla f(\mathbf {x} )\cdot \mathbf {u} }
36845:
36613:
36495:
36316:
36264:
36070:
36048:
35916:
34550:
23383:{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {g(X)} )}{\partial \mathbf {X} }}=}
18793:
10232:
4973:
3192:
2254:
2174:
2001:
Matrix calculus is used for deriving optimal stochastic estimators, often involving the use of
1050:
1011:
1007:
995:
966:
895:
856:
740:
676:
600:
33915:
28814:{\displaystyle -\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\mathbf {U} ^{-1}}
21344:{\displaystyle {\frac {\partial g(u)}{\partial u}}{\frac {\partial u}{\partial \mathbf {X} }}}
13103:{\displaystyle {\frac {\partial g(u)}{\partial u}}{\frac {\partial u}{\partial \mathbf {x} }}}
36739:
36598:
36510:
36167:
36102:
36075:
36065:
35986:
35974:
35959:
35931:
35708:
23168:{\displaystyle {\frac {\partial \operatorname {tr} (a\mathbf {U} )}{\partial \mathbf {X} }}=}
14292:{\displaystyle {\frac {\partial ^{2}f}{\partial \mathbf {x} \partial \mathbf {x} ^{\top }}}=}
2315:. The corresponding concept from vector calculus is indicated at the end of each subsection.
2245:
2035:
The vector and matrix derivatives presented in the sections to follow take full advantage of
1035:
1023:
1015:
940:
606:
382:
327:
288:
194:
35751:
35313:
23225:{\displaystyle a{\frac {\partial \operatorname {tr} (\mathbf {U} )}{\partial \mathbf {X} }}}
22863:{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {X} )}{\partial \mathbf {X} }}=}
16747:{\displaystyle {\frac {(\mathbf {x} -\mathbf {a} )^{\top }}{\|\mathbf {x} -\mathbf {a} \|}}}
16676:{\displaystyle {\frac {\partial \;\|\mathbf {x} -\mathbf {a} \|}{\partial \;\mathbf {x} }}=}
1981:
36555:
36174:
36021:
35512:
Liu, Shuangzhe; Leiva, Victor; Zhuang, Dan; Ma, Tiefeng; Figueroa-Zúñiga, Jorge I. (2022).
35379:
34512:
10853:{\displaystyle {\frac {\partial \mathbf {x} ^{\top }\mathbf {A} }{\partial \mathbf {x} }}=}
10215:
To help make sense of all the identities below, keep in mind the most important rules: the
5190:
It is the gradient matrix, in particular, that finds many uses in minimization problems in
2729:
2434:
2002:
1373:
1046:
999:
945:
925:
851:
520:
444:
418:
332:
33660:
15776:{\displaystyle \mathbf {a} ^{\top }{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}
11627:{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}\mathbf {A} ^{\top }}
8:
36575:
36500:
36387:
36344:
36095:
36080:
35911:
35899:
35886:
35846:
35826:
34504:
32356:
17639:{\displaystyle {\frac {\partial (\mathbf {u} ^{\top }\times \mathbf {v} )}{\partial x}}=}
5195:
1985:
920:
890:
880:
767:
621:
423:
279:
162:
157:
31019:{\displaystyle \operatorname {tr} \left(\mathbf {A} \mathbf {g} '(x\mathbf {A} )\right)}
26132:{\displaystyle {\frac {\partial \left|\mathbf {X} ^{n}\right|}{\partial \mathbf {X} }}=}
18155:{\displaystyle {\frac {\partial \mathbf {f} (\mathbf {g} (\mathbf {u} ))}{\partial x}}=}
6735:×1 column vector, then the result using the numerator layout will be in the form of a 1×
5223:
The fundamental issue is that the derivative of a vector with respect to a vector, i.e.
3218:(represented in this case as a column vector) is defined using the gradient as follows.
36664:
36639:
36470:
36423:
36364:
36329:
36324:
36304:
36299:
36294:
36259:
36206:
36189:
36090:
35964:
35949:
35894:
35861:
35558:
34532:
32020:{\displaystyle (d\mathbf {X} )\otimes \mathbf {Y} +\mathbf {X} \otimes (d\mathbf {Y} )}
31625:
24658:{\displaystyle \mathbf {BX^{\top }CA} +\mathbf {B^{\top }X^{\top }A^{\top }C^{\top }} }
6724:
3185:
2249:
2215:
1990:
1003:
885:
788:
772:
712:
707:
702:
666:
547:
471:
377:
372:
176:
171:
31:
32731:{\displaystyle \int _{0}^{1}e^{a\mathbf {X} }(d\mathbf {X} )e^{(1-a)\mathbf {X} }\,da}
11719:{\displaystyle {\frac {\partial (\mathbf {u} +\mathbf {v} )}{\partial \mathbf {x} }}=}
36804:
36628:
36560:
36382:
36359:
36233:
36226:
36129:
35944:
35836:
35613:
35596:
35586:
35562:
35492:
35467:
35442:
35413:
35365:
34555:
34526:
31915:
11862:{\displaystyle {\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {x} }}=}
6745:
Result of differentiating various kinds of aggregates with other kinds of aggregates
5191:
2709:
1058:
959:
793:
571:
454:
407:
264:
259:
35338:
32587:{\displaystyle \sum _{i=0}^{n-1}\mathbf {X} ^{i}(d\mathbf {X} )\mathbf {X} ^{n-i-1}}
26259:{\displaystyle n\left|\mathbf {X} ^{n}\right|\left(\mathbf {X} ^{-1}\right)^{\top }}
25616:{\displaystyle \operatorname {cofactor} (X)^{\top }=|\mathbf {X} |\mathbf {X} ^{-1}}
18768:{\displaystyle {\frac {\partial \mathbf {f} (\mathbf {g} )}{\partial \mathbf {g} }}}
18716:{\displaystyle {\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}}
36762:
36545:
36458:
36438:
36369:
36279:
36221:
36213:
36147:
36060:
35821:
35816:
35763:
35754:(notes on matrix differentiation), Pawel Koval, from Munich Personal RePEc Archive.
35636:
35550:
35525:
35357:
34545:
33495:{\displaystyle (\mathbf {P} _{k})_{ij}=(\mathbf {Q} )_{ik}(\mathbf {Q} ^{-1})_{kj}}
32137:{\displaystyle (d\mathbf {X} )\circ \mathbf {Y} +\mathbf {X} \circ (d\mathbf {Y} )}
28011:{\displaystyle \mathbf {A} {\frac {\partial \mathbf {U} }{\partial x}}\mathbf {B} }
15284:
14991:
12400:{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}=\nabla _{\mathbf {x} }y}
11526:{\displaystyle {\frac {\partial \mathbf {A} \mathbf {u} }{\partial \mathbf {x} }}=}
10728:{\displaystyle {\frac {\partial \mathbf {A} \mathbf {x} }{\partial \mathbf {x} }}=}
6711:
4145:
for pedagogical purposes. Some authors use different conventions. The section on
2342:
2325:
for pedagogical purposes. Some authors use different conventions. The section on
2024:
1062:
803:
697:
671:
532:
449:
413:
33565:{\displaystyle \mathbf {X} =\mathbf {Q} {\boldsymbol {\Lambda }}\mathbf {Q} ^{-1}}
28418:{\displaystyle {\frac {\partial (\mathbf {U} \otimes \mathbf {V} )}{\partial x}}=}
23986:{\displaystyle \mathbf {X} ^{\top }\left(\mathbf {A} +\mathbf {A} ^{\top }\right)}
16805:{\displaystyle {\frac {\mathbf {x} -\mathbf {a} }{\|\mathbf {x} -\mathbf {a} \|}}}
15866:{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}\mathbf {a} }
14909:{\displaystyle \mathbf {x} ^{\top }\left(\mathbf {A} +\mathbf {A} ^{\top }\right)}
11573:{\displaystyle \mathbf {A} {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}
36824:
36809:
36593:
36448:
36428:
36397:
36374:
36248:
35904:
35851:
35731:
35375:
33399:
25077:{\displaystyle \sum _{i=0}^{n-1}\mathbf {X} ^{i}\mathbf {A} \mathbf {X} ^{n-i-1}}
18470:{\displaystyle {\frac {\partial (\mathbf {U} \times \mathbf {v} )}{\partial x}}=}
5073:
4984:
4020:
3997:
3379:
2994:
2701:
2610:
2294:
2275:
2266:
2036:
1429:. This type of generalized derivative can be seen as the derivative of a scalar,
1090:
935:
808:
762:
644:
557:
502:
35:
35361:
29759:{\displaystyle {\frac {\partial (\mathbf {u} \cdot \mathbf {v} )}{\partial x}}=}
28589:{\displaystyle {\frac {\partial (\mathbf {U} \circ \mathbf {V} )}{\partial x}}=}
17387:{\displaystyle \left({\frac {\partial \mathbf {u} }{\partial x}}\right)^{\top }}
5330:
laid out in columns, or vice versa. This leads to the following possibilities:
36734:
36633:
36480:
36433:
36334:
36137:
35640:
35554:
35530:
35513:
35130:
28902:{\displaystyle {\frac {\partial ^{2}\mathbf {U} ^{-1}}{\partial x\partial y}}=}
26435:
26271:
17264:{\displaystyle {\frac {\partial \mathbf {u} }{\partial x}}\mathbf {A} ^{\top }}
14353:
11207:{\displaystyle {\frac {\partial v}{\partial \mathbf {x} }}\mathbf {a} ^{\top }}
5758:
we have the same issues. To be consistent, we should do one of the following:
3177:
2705:
2627:
818:
626:
398:
36152:
31544:{\displaystyle \operatorname {tr} \left(\mathbf {X} ^{-1}d\mathbf {X} \right)}
25776:{\displaystyle {\frac {\partial \ln |a\mathbf {X} |}{\partial \mathbf {X} }}=}
2214:: As mentioned above, there are competing notations for laying out systems of
36839:
36608:
36463:
36349:
36053:
36028:
35714:
35711:, Randal J. Barnes, Department of Civil Engineering, University of Minnesota.
35600:
35489:
Matrix differential calculus with applications in statistics and econometrics
34602:
33881:{\displaystyle \mathbf {P} _{i}\mathbf {P} _{j}=\delta _{ij}\mathbf {P} _{i}}
33034:{\displaystyle \mathbf {P} _{i}\mathbf {P} _{j}=\delta _{ij}\mathbf {P} _{i}}
32344:{\displaystyle -\mathbf {X} ^{-1}\left(d\mathbf {X} \right)\mathbf {X} ^{-1}}
31155:{\displaystyle \operatorname {tr} \left(\mathbf {A} e^{x\mathbf {A} }\right)}
29273:{\displaystyle {\frac {\partial \,\mathbf {g} (x\mathbf {A} )}{\partial x}}=}
23281:
7167:
6913:
5615:
A third possibility sometimes seen is to insist on writing the derivative as
5199:
2109:
2014:
2009:
798:
562:
317:
274:
35608:
Lax, Peter D. (2007). "9. Calculus of Vector- and Matrix-Valued
Functions".
36618:
36588:
36453:
36016:
35745:
35405:
29481:{\displaystyle \mathbf {A} e^{x\mathbf {A} }=e^{x\mathbf {A} }\mathbf {A} }
28101:{\displaystyle {\frac {\partial (\mathbf {U} +\mathbf {V} )}{\partial x}}=}
25940:{\displaystyle {\frac {\partial |\mathbf {AXB} |}{\partial \mathbf {X} }}=}
18784:
17477:{\displaystyle {\frac {\partial (\mathbf {u} +\mathbf {v} )}{\partial x}}=}
10220:
2713:
1054:
552:
302:
28253:{\displaystyle {\frac {\partial (\mathbf {U} \mathbf {V} )}{\partial x}}=}
24039:{\displaystyle \left(\mathbf {A} +\mathbf {A} ^{\top }\right)\mathbf {X} }
17917:{\displaystyle {\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial x}}=}
14962:{\displaystyle \left(\mathbf {A} +\mathbf {A} ^{\top }\right)\mathbf {x} }
10986:{\displaystyle {\frac {\partial a\mathbf {u} }{\partial \,\mathbf {x} }}=}
8480:
The following definitions are only provided in numerator-layout notation:
35866:
35808:
35671:
35434:
29526:
25543:{\displaystyle {\frac {\partial |\mathbf {X} |}{\partial \mathbf {X} }}=}
4977:
2196:
2055:
1031:
987:
915:
17167:{\displaystyle {\frac {\partial \mathbf {A} \mathbf {u} }{\partial x}}=}
11288:{\displaystyle {\frac {\partial v\mathbf {u} }{\partial \mathbf {x} }}=}
11110:{\displaystyle {\frac {\partial v\mathbf {a} }{\partial \mathbf {x} }}=}
5657:{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} '}},}
36583:
36515:
36269:
36142:
36006:
35996:
35939:
31903:{\displaystyle (d\mathbf {X} )\mathbf {Y} +\mathbf {X} (d\mathbf {Y} )}
24728:{\displaystyle \mathbf {A^{\top }C^{\top }XB^{\top }} +\mathbf {CAXB} }
23247:
18788:
17212:{\displaystyle \mathbf {A} {\frac {\partial \mathbf {u} }{\partial x}}}
11155:{\displaystyle \mathbf {a} {\frac {\partial v}{\partial \mathbf {x} }}}
11031:{\displaystyle a{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}
10640:{\displaystyle {\frac {\partial \mathbf {x} }{\partial \mathbf {x} }}=}
10568:{\displaystyle {\frac {\partial \mathbf {a} }{\partial \mathbf {x} }}=}
10216:
7055:
6979:
6146:{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}.}
5927:{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }},}
5799:{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }},}
3363:
2725:
2338:
1027:
661:
585:
312:
307:
211:
35627:
Magnus, Jan R. (October 2010). "On the concept of matrix derivative".
30286:{\displaystyle {\frac {\partial ^{2}|\mathbf {U} |}{\partial x^{2}}}=}
26063:{\displaystyle |\mathbf {AXB} |\left(\mathbf {X} ^{-1}\right)^{\top }}
15907:{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}
15817:{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}
10472:{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}
7338:{\displaystyle {\frac {\partial \mathbf {Y} }{\partial \mathbf {X} }}}
7294:{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {X} }}}
7147:{\displaystyle {\frac {\partial \mathbf {Y} }{\partial \mathbf {x} }}}
7093:{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}
5599:{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}
5466:{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}
5403:{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}
5257:{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}
1896:{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}
36777:
36525:
36520:
35831:
35134:
33754:{\textstyle f(\mathbf {X} )=\sum _{i}f(\lambda _{i})\mathbf {P} _{i}}
2274:
The notations developed here can accommodate the usual operations of
2162:
590:
580:
33502:
is the set of orthogonal projection operators that project onto the
21602:{\displaystyle {\frac {\partial g(\mathbf {U} )}{\partial X_{ij}}}=}
17332:{\displaystyle {\frac {\partial \mathbf {u} ^{\top }}{\partial x}}=}
6555:
In the following formulas, we handle the five possible combinations
2079:
columns. Such matrices will be denoted using bold capital letters:
36772:
36274:
35800:
35664:, a website for evaluating matrix calculus expressions symbolically
34378:{\displaystyle {\frac {d\mathbf {y} }{d\mathbf {x} }}=\mathbf {A} }
34144:{\displaystyle dy=\operatorname {tr} (\mathbf {A} \,d\mathbf {X} )}
5935:
5807:
5675:
5565:
layout in the previous example, which means that the row number of
5515:
5369:
layout in the previous example, which means that the row number of
3181:
2998:
2697:
1086:
1079:
656:
403:
360:
49:
32960:{\displaystyle \mathbf {X} =\sum _{i}\lambda _{i}\mathbf {P} _{i}}
30130:{\displaystyle {\frac {\partial \ln |\mathbf {U} |}{\partial x}}=}
21423:{\displaystyle {\frac {\partial f(g(u))}{\partial \mathbf {X} }}=}
13182:{\displaystyle {\frac {\partial f(g(u))}{\partial \mathbf {x} }}=}
12344:
The fundamental identities are placed above the thick black line.
1670:
matrix consisting of all of the possible derivative combinations.
36623:
35876:
35661:
29420:{\displaystyle {\frac {\partial e^{x\mathbf {A} }}{\partial x}}=}
28741:{\displaystyle {\frac {\partial \mathbf {U} ^{-1}}{\partial x}}=}
21875:{\displaystyle {\frac {\partial \mathbf {U} }{\partial X_{ij}}},}
1039:
35667:
35585:. Econometric Exercises. Cambridge: Cambridge University Press.
34089:{\displaystyle {\frac {dy}{d\mathbf {x} }}=\mathbf {a} ^{\top }}
23478:{\displaystyle \left(\mathbf {g} '(\mathbf {X} )\right)^{\top }}
20990:{\displaystyle {\frac {\partial (u+v)}{\partial \mathbf {X} }}=}
12749:{\displaystyle {\frac {\partial (u+v)}{\partial \mathbf {x} }}=}
36792:
35856:
33811:{\textstyle \mathbf {X} =\sum _{i}\lambda _{i}\mathbf {P} _{i}}
30598:{\displaystyle {\frac {\partial g(\mathbf {U} )}{\partial x}}=}
29593:{\displaystyle {\frac {\partial g(\mathbf {u} )}{\partial x}}=}
26188:{\displaystyle n\left|\mathbf {X} ^{n}\right|\mathbf {X} ^{-1}}
24181:{\displaystyle -\mathbf {X} ^{-1}\mathbf {A} \mathbf {X} ^{-1}}
21275:{\displaystyle {\frac {\partial g(u)}{\partial \mathbf {X} }}=}
13034:{\displaystyle {\frac {\partial g(u)}{\partial \mathbf {x} }}=}
5664:(i.e. the derivative is taken with respect to the transpose of
4972:
Important examples of scalar functions of matrices include the
1964:
1674:
organized in matrix form are collected in the following table.
29967:{\displaystyle {\frac {\partial |\mathbf {U} |}{\partial x}}=}
27961:{\displaystyle {\frac {\partial \mathbf {AUB} }{\partial x}}=}
3021:) is the transpose of the derivative of a scalar by a vector.
35871:
35677:
34674:
disappears in the result. This is intentional. In general,
27834:{\displaystyle {\frac {\partial a\mathbf {U} }{\partial x}}=}
21134:{\displaystyle {\frac {\partial uv}{\partial \mathbf {X} }}=}
20859:{\displaystyle {\frac {\partial au}{\partial \mathbf {X} }}=}
17047:{\displaystyle {\frac {\partial a\mathbf {u} }{\partial x}}=}
12893:{\displaystyle {\frac {\partial uv}{\partial \mathbf {x} }}=}
12618:{\displaystyle {\frac {\partial au}{\partial \mathbf {x} }}=}
6544:{\displaystyle {\frac {\partial y}{\partial \mathbf {X} '}},}
35296:{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}.}
35248:{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}},}
35194:
33616:{\displaystyle ({\boldsymbol {\Lambda }})_{ii}=\lambda _{i}}
27877:{\displaystyle a{\frac {\partial \mathbf {U} }{\partial x}}}
20902:{\displaystyle a{\frac {\partial u}{\partial \mathbf {X} }}}
20730:{\displaystyle {\frac {\partial a}{\partial \mathbf {X} }}=}
17090:{\displaystyle a{\frac {\partial \mathbf {u} }{\partial x}}}
16947:{\displaystyle {\frac {\partial \mathbf {a} }{\partial x}}=}
12661:{\displaystyle a{\frac {\partial u}{\partial \mathbf {x} }}}
12488:{\displaystyle {\frac {\partial a}{\partial \mathbf {x} }}=}
6295:{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}},}
6260:. In practice, however, following a denominator layout for
6248:, while consistent denominator layout lays out according to
6229:{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}},}
6097:{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}},}
6055:{\displaystyle {\frac {\partial y}{\partial \mathbf {x} '}}}
5751:{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}},}
5554:{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }},}
5264:, is often written in two competing ways. If the numerator
35769:
33347:
27754:{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}}
20648:{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}}
16859:{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}}
7240:{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}}
7042:{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}}
6949:{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}}
6900:{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}}
6703:{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}}
6488:{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}}
6443:{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}}
6392:{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}}
6347:{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}}
6187:{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}}
6008:{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}}
5969:{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}}
5880:{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}}
5841:{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}}
5709:{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}}
2669:{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}}
2226:
in more detail. It is important to realize the following:
1943:{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}}
1854:{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}}
1809:{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}}
1769:{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}}
35110:"Appendix D, Linear Algebra: Determinants, Inverses, Rank"
32399:{\displaystyle d\left(\mathbf {X} ^{\mathrm {H} }\right)=}
18671:: The formulas involving the vector-by-vector derivatives
10238:
The following identities adopt the following conventions:
35544:
34328:{\displaystyle d\mathbf {y} =\mathbf {A} \,d\mathbf {x} }
6155:
Similarly, when it comes to scalar-by-matrix derivatives
1455:, and its result can be easily collected in vector form.
35412:. Science Press (Beijing) and Springer-Verlag (Berlin).
34469:{\displaystyle {\frac {d\mathbf {Y} }{dx}}=\mathbf {A} }
34283:{\displaystyle {\frac {d\mathbf {y} }{dx}}=\mathbf {a} }
34192:{\displaystyle {\frac {dy}{d\mathbf {X} }}=\mathbf {A} }
24896:{\displaystyle n\left(\mathbf {X} ^{n-1}\right)^{\top }}
1093:. For a scalar function of three independent variables,
6236:
then consistent numerator layout lays out according to
2020:
Expectation-maximization algorithm for
Gaussian mixture
33767:
33692:
26423:{\displaystyle 2\left(\mathbf {X} ^{+}\right)^{\top }}
25857:{\displaystyle \left(\mathbf {X} ^{-1}\right)^{\top }}
9861:
9446:
9302:
9154:
8881:
8528:
8134:
7719:
7569:
7427:
5513:(numerator layout), which is its transpose. (However,
5202:
algorithm, which is of great importance in the field.
4630:
4217:
3606:
3492:
3403:
3045:
2870:
2758:
2486:
2365:
1522:
35267:
35219:
34879:
34680:
34637:
34585:
34435:
34394:
34342:
34299:
34249:
34208:
34158:
34105:
34048:
34001:
33955:
33918:
33825:
33663:
33629:
33578:
33528:
33408:
33374:
33134:
33086:
33057:
32978:
32916:
32794:
32749:
32649:
32605:
32506:
32462:
32413:
32367:
32289:
32242:
32199:
32155:
32086:
32043:
31969:
31926:
31858:
31818:
31777:
31734:
31700:
31662:
31628:
31593:
31501:
31450:
31336:
31291:
31251:
31237:{\displaystyle d(\operatorname {tr} (\mathbf {X} ))=}
31204:
31115:
31048:
30970:
30901:
30701:
30612:
30560:
30300:
30233:
30144:
30085:
29981:
29928:
29862:
Identities: scalar-by-scalar, with matrices involved
29773:
29716:
29607:
29555:
29434:
29381:
29287:
29229:
28916:
28850:
28755:
28702:
28603:
28546:
28432:
28375:
28267:
28213:
28115:
28058:
27975:
27926:
27848:
27802:
27728:
27528:
27328:
27256:
27129:
27011:
26939:
26826:
26687:
26615:
26556:
26527:
26447:
26389:
26356:
26283:
26202:
26146:
26086:
26008:
25954:
25893:
25823:
25790:
25726:
25630:
25557:
25502:
25449:
25414:
25345:
25326:{\displaystyle \left(e^{\mathbf {X} }\right)^{\top }}
25295:
25265:
25199:
25091:
25005:
24932:
24856:
24817:
24751:
24672:
24588:
24510:
24443:
24417:
24309:
24195:
24139:
24069:
24000:
23940:
23868:
23829:
23799:
23679:
23633:
23610:
23508:
23436:
23397:
23331:
23182:
23122:
22998:
22933:
22877:
22820:
22734:
22635:
22551:
22440:
22339:
22239:
22177:
22142:
22073:
22017:
21982:
21920:
21836:
21715:
21616:
21554:
21437:
21376:
21289:
21237:
21148:
21102:
21004:
20949:
20873:
20827:
20774:
20744:
20701:
20622:
20476:
20328:
19078:
19004:
18805:
18729:
18677:
18565:
18484:
18427:
18279:
18169:
18104:
18006:
17931:
17877:
17756:
17653:
17589:
17491:
17434:
17346:
17296:
17226:
17181:
17133:
17061:
17015:
16961:
16918:
16833:
16761:
16690:
16629:
16484:
16354:
16236:
16117:
16030:
15944:
15879:
15833:
15789:
15736:
15638:
15581:
15548:
15408:
15378:
15294:
15231:
15147:
15101:
15063:
15001:
14923:
14863:
14801:
14751:
14716:
14654:
14604:
14574:
14478:
14378:
14336:
14306:
14241:
14166:
14071:
14000:
13891:
13783:
13666:
13588:
13517:
13425:
13327:
13196:
13135:
13048:
12996:
12907:
12861:
12763:
12708:
12632:
12586:
12532:
12502:
12459:
12356:
12227:
12115:
12048:
11953:
11876:
11820:
11733:
11674:
11587:
11540:
11490:
11377:
11302:
11254:
11169:
11124:
11076:
11000:
10951:
10897:
10867:
10810:
10765:
10742:
10692:
10654:
10609:
10582:
10537:
10444:
9115:
8489:
7388:
7310:
7266:
7214:
7119:
7065:
7016:
6923:
6874:
6833:
6677:
6561:
6551:
with results the same as consistent numerator layout.
6510:
6462:
6417:
6366:
6321:
6266:
6200:
6161:
6115:
6068:
6024:
5982:
5943:
5896:
5854:
5815:
5768:
5722:
5683:
5621:
5571:
5525:
5438:
5416:
5375:
5229:
5112:
4996:
4599:
4186:
4051:
3573:
3477:
3388:
3303:
3227:
3030:
3015:(whose independent coordinates are the components of
2941:
2905:
2874:
2839:
2743:
2712:
vector (considered as a function of time). Also, the
2643:
2455:
2350:
2137:
is a scalar, denoted with lowercase italic typeface:
1917:
1868:
1828:
1783:
1743:
1705:
1588:
1557:
1526:
1464:
1439:
1403:
1376:
1340:
1164:
1099:
62:
35744:(notes on matrix differentiation, in the context of
34653:
refers to a matrix of all 0's, of the same shape as
34522:
34034:{\displaystyle dy=\mathbf {a} ^{\top }d\mathbf {x} }
31955:{\displaystyle d(\mathbf {X} \otimes \mathbf {Y} )=}
29514:
Identities: scalar-by-scalar, with vectors involved
20607:
35674:
package that has some matrix calculus functionality
35612:(2nd ed.). Hoboken, N.J.: Wiley-Interscience.
35184:Petersen, Kaare Brandt; Pedersen, Michael Syskind.
32185:{\displaystyle d\left(\mathbf {X} ^{\top }\right)=}
4987:this derivative is often written as the following.
4012:whose components represent a space is known as the
35295:
35247:
35179:
35177:
35175:
35173:
35171:
35169:
35167:
35087:"Old and New Matrix Algebra Useful for Statistics"
35048:
34865:
34645:
34593:
34468:
34420:
34377:
34327:
34282:
34234:
34191:
34143:
34088:
34033:
33984:
33940:
33880:
33810:
33753:
33678:
33646:
33615:
33564:
33494:
33390:
33353:
33119:
33070:
33033:
32959:
32899:
32779:
32730:
32634:
32586:
32491:
32439:
32398:
32343:
32274:
32223:
32184:
32136:
32072:{\displaystyle d(\mathbf {X} \circ \mathbf {Y} )=}
32071:
32019:
31954:
31902:
31843:
31799:
31762:
31715:
31685:
31634:
31613:
31543:
31486:
31433:
31321:
31275:{\displaystyle \operatorname {tr} (d\mathbf {X} )}
31274:
31236:
31169:
31154:
31100:
31018:
30955:
30790:
30686:
30597:
30527:
30285:
30200:
30129:
30052:
29966:
29839:
29758:
29668:
29592:
29480:
29419:
29351:
29272:
29118:
28901:
28813:
28740:
28669:
28588:
28498:
28417:
28327:
28252:
28165:
28100:
28010:
27960:
27876:
27833:
27753:
27699:
27513:
27313:
27211:
27114:
26996:
26894:
26811:
26672:
26577:
26541:
26512:
26422:
26374:
26341:
26258:
26187:
26131:
26062:
25993:
25939:
25856:
25808:
25775:
25696:
25615:
25542:
25482:
25434:
25399:
25325:
25280:
25250:
25179:
25076:
24990:
24895:
24841:
24802:
24727:
24657:
24573:
24468:
24428:
24402:
24273:
24180:
24124:
24038:
23985:
23925:
23837:
23814:
23784:
23648:
23618:
23595:
23477:
23421:
23382:
23224:
23167:
23076:
22983:
22885:
22862:
22801:
22719:
22620:
22509:
22425:
22324:
22197:
22162:
22127:
22037:
22002:
21967:
21874:
21814:
21700:
21601:
21520:
21422:
21343:
21274:
21204:
21133:
21054:
20989:
20901:
20858:
20782:
20759:
20729:
20647:
20594:
20460:
20309:
19065:
18987:
18767:
18715:
18653:
18550:
18469:
18374:
18264:
18154:
18066:
17991:
17916:
17844:
17741:
17638:
17541:
17476:
17386:
17331:
17263:
17211:
17166:
17089:
17046:
16969:
16946:
16858:
16804:
16746:
16675:
16599:
16469:
16339:
16182:
16102:
16015:
15906:
15865:
15816:
15775:
15721:
15592:
15566:
15533:
15389:
15363:
15254:
15216:
15117:
15086:
15048:
14961:
14908:
14848:
14771:
14736:
14701:
14612:
14589:
14559:
14463:
14344:
14321:
14291:
14220:
14152:
14054:
13986:
13876:
13720:
13652:
13571:
13503:
13410:
13279:
13181:
13102:
13033:
12963:
12892:
12813:
12748:
12660:
12617:
12540:
12517:
12487:
12399:
12324:
12212:
12100:
12015:
11938:
11861:
11787:
11718:
11626:
11572:
11525:
11444:
11362:
11287:
11206:
11154:
11109:
11030:
10985:
10905:
10882:
10852:
10780:
10750:
10727:
10662:
10639:
10590:
10567:
10471:
10196:
9090:
8469:
7337:
7293:
7239:
7146:
7092:
7041:
6948:
6899:
6856:
6702:
6663:
6543:
6487:
6442:
6391:
6346:
6294:
6228:
6186:
6145:
6096:
6054:
6007:
5968:
5926:
5879:
5840:
5798:
5750:
5708:
5656:
5598:
5553:
5465:
5424:
5402:
5256:
5179:
5061:
4961:
4548:
4111:
3985:
3552:
3463:
3354:
3286:
3165:
2982:
2818:
2668:
2598:
2425:
1942:
1895:
1853:
1808:
1768:
1728:
1637:
1447:
1421:
1389:
1362:
1323:
1147:
134:
35742:Introduction to Vector and Matrix Differentiation
35165:
35163:
35161:
35159:
35157:
35155:
35153:
35151:
35149:
35147:
35117:ASEN 5007: Introduction To Finite Element Methods
31186:Differential identities: scalar involving matrix
27688:
27567:
27502:
27367:
25994:{\displaystyle |\mathbf {AXB} |\mathbf {X} ^{-1}}
23357:
23351:
15255:{\displaystyle \mathbf {A} +\mathbf {A} ^{\top }}
15087:{\displaystyle 2\mathbf {x} ^{\top }\mathbf {A} }
1656:-vector of dependent variables, or functions, of
36837:
35684:, as well as symbolic tensor derivatives in its
35511:
35314:"Properties of the Trace and Matrix Derivatives"
33900:Conversion from differential to derivative form
32275:{\displaystyle d\left(\mathbf {X} ^{-1}\right)=}
22198:{\displaystyle \mathbf {b} \mathbf {a} ^{\top }}
22163:{\displaystyle \mathbf {a} \mathbf {b} ^{\top }}
22038:{\displaystyle \mathbf {a} \mathbf {b} ^{\top }}
22003:{\displaystyle \mathbf {b} \mathbf {a} ^{\top }}
14772:{\displaystyle \mathbf {A} ^{\top }\mathbf {b} }
14737:{\displaystyle \mathbf {b} ^{\top }\mathbf {A} }
6857:{\displaystyle {\frac {\partial y}{\partial x}}}
5561:regardless of layout.). This corresponds to the
1975:
1729:{\displaystyle {\frac {\partial y}{\partial x}}}
135:{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}
35486:
35439:Growth curve models and statistical diagnostics
32635:{\displaystyle d\left(e^{\mathbf {X} }\right)=}
32492:{\displaystyle d\left(\mathbf {X} ^{n}\right)=}
20608:Conversion from differential to derivative form
5286:, then the result can be laid out as either an
2112:, is denoted with a boldface lowercase letter:
1155:, the gradient is given by the vector equation
27:Specialized notation for multivariable calculus
35766:(notes on matrix differentiation), Sam Roweis.
35464:Advanced multivariate statistics with matrices
35461:
35144:
35141:) definition of vector and matrix derivatives.
34421:{\displaystyle d\mathbf {Y} =\mathbf {A} \,dx}
34235:{\displaystyle d\mathbf {y} =\mathbf {a} \,dx}
33908:Equivalent derivative form (numerator layout)
33120:{\displaystyle d\left(f(\mathbf {X} )\right)=}
32440:{\displaystyle (d\mathbf {X} )^{\mathrm {H} }}
30528:{\displaystyle \left|\mathbf {U} \right|\left}
29501:
27714:
18778:
16819:
12339:
10426:
10394:are constant in respect of, and the matrices,
9101:
4135:respectively after their analogs for vectors.
1649:
35785:
35709:Matrix Differentiation (and some other stuff)
31763:{\displaystyle d(\mathbf {X} +\mathbf {Y} )=}
24469:{\displaystyle \mathbf {A^{\top }B^{\top }} }
18386:Assumes consistent matrix layout; see below.
18078:Assumes consistent matrix layout; see below.
10333:are constant in respect of, and the vectors,
10272:are constant in respect of, and the scalars,
4141:: The discussion in this section assumes the
2321:: The discussion in this section assumes the
1026:. The notation used here is commonly used in
967:
35680:supports symbolic matrix derivatives in its
35580:
35183:
34627:
34625:
31844:{\displaystyle d(\mathbf {X} \mathbf {Y} )=}
25483:{\displaystyle (\cos(\mathbf {X} ))^{\top }}
21884:i.e. mixed layout if denominator layout for
16796:
16780:
16738:
16722:
16653:
16637:
9106:Using denominator-layout notation, we have:
7374:
4121:
35119:. Boulder, Colorado: University of Colorado
31800:{\displaystyle d\mathbf {X} +d\mathbf {Y} }
23422:{\displaystyle \mathbf {g} '(\mathbf {X} )}
5205:
2260:
35792:
35778:
35089:. MIT Media Lab note (1997; revised 12/00)
18783:Note that exact equivalents of the scalar
16661:
16636:
16323:
16243:
15999:
15951:
7379:Using numerator-layout notation, we have:
6731:using matrix calculus, if the domain is a
3382:(a vector whose components are functions)
974:
960:
36820:Regiomontanus' angle maximization problem
35581:Abadir, Karim M.; Magnus, Jan R. (2005).
35529:
35462:Kollo, Tõnu; von Rosen, Dietrich (2005).
35404:
35084:
34622:
34575:
34573:
34571:
34509:ordinary least squares regression formula
34411:
34316:
34225:
34129:
33931:
33623:are the eigenvalues. The matrix function
32890:
32871:
32824:
32721:
31704:
29854:
29236:
15118:{\displaystyle 2\mathbf {A} \mathbf {x} }
10971:
6018:In the third possibility above, we write
1626:
95:
36663:
29506:
4027:The pushforward along a vector function
1018:with respect to a single variable, into
36168:Differentiating under the integral sign
35432:
35207:This book uses a mixed layout, i.e. by
35080:
35078:
35076:
35074:
35072:
33656:defined in terms of the scalar function
32780:{\displaystyle d\left(\log {X}\right)=}
32224:{\displaystyle (d\mathbf {X} )^{\top }}
6302:and laying the result out according to
4587:
4586:of independent variables, is given (in
4174:
4146:
4142:
3561:
2827:
2716:is the tangent vector of the velocity.
2443:
2326:
2322:
2223:
2219:
1969:
1066:
498:Differentiating under the integral sign
14:
36838:
35626:
35487:Magnus, Jan; Neudecker, Heinz (2019).
35085:Thomas P., Minka (December 28, 2000).
34568:
31487:{\displaystyle d(\ln |\mathbf {X} |)=}
26578:{\displaystyle -2\mathbf {X} ^{\top }}
4000:, the derivative of a vector function
3544:
3455:
3157:
2810:
2417:
1984:is the standard way in the setting of
1507:
36044:Inverse functions and differentiation
35773:
35760:More notes on matrix differentiation.
35354:Advances in Automatic Differentiation
35350:
30893:is the corresponding matrix function.
29888:
29872:
29869:
29866:
29524:
29518:
27770:
27767:
27764:
20672:
20664:
20658:
16886:
16875:
16872:
16869:
15567:{\displaystyle 2\mathbf {x} ^{\top }}
12427:
12416:
12413:
12410:
10502:
10488:
10482:
6749:
5505:. Some authors term this layout the
5214:
2199:. All functions are assumed to be of
35069:
29521:
29221:is the corresponding matrix function
23815:{\displaystyle \mathbf {A} ^{\top }}
23649:{\displaystyle \mathbf {A} ^{\top }}
23323:is the corresponding matrix function
20760:{\displaystyle \mathbf {0} ^{\top }}
20661:
14590:{\displaystyle \mathbf {a} ^{\top }}
14322:{\displaystyle \mathbf {H} ^{\top }}
12518:{\displaystyle \mathbf {0} ^{\top }}
10883:{\displaystyle \mathbf {A} ^{\top }}
10781:{\displaystyle \mathbf {A} ^{\top }}
10485:
5890:If we choose denominator layout for
5410:equals to the size of the numerator
4564:The derivative of a scalar function
4157:The derivative of a matrix function
3212:in the direction of the unit vector
2696:Simple examples of this include the
1148:{\displaystyle f(x_{1},x_{2},x_{3})}
994:is a specialized notation for doing
35610:Linear algebra and its applications
35607:
35107:
30865:is the equivalent scalar function,
29193:is the equivalent scalar function,
24842:{\displaystyle n\mathbf {X} ^{n-1}}
23295:is the equivalent scalar function,
16589:
16579:
16572:
16562:
16549:
16535:
16525:
16518:
16502:
16488:
16462:
16455:
16438:
16428:
16421:
16408:
16395:
16377:
16367:
16360:
16326:
16310:
16300:
16293:
16283:
16266:
16256:
16249:
16175:
16157:
16149:
16133:
16125:
16084:
16076:
16060:
16052:
16034:
16002:
15989:
15976:
15968:
15955:
5519:more commonly means the derivative
5501:). This is sometimes known as the
5355:). This is sometimes known as the
4559:
4152:
3471:, with respect to an input vector,
3369:
2719:
2332:
2005:. This includes the derivation of:
1972:section for a more detailed table.
24:
35842:Free variables and bound variables
35752:A note on differentiating matrices
35573:
35279:
35271:
35233:
35223:
34081:
34018:
33985:{\displaystyle {\frac {dy}{dx}}=a}
32805:
32431:
32383:
32216:
32170:
31322:{\displaystyle d(|\mathbf {X} |)=}
31086:
31052:
30941:
30905:
30774:
30764:
30756:
30740:
30721:
30670:
30660:
30646:
30627:
30583:
30564:
30496:
30486:
30435:
30425:
30365:
30349:
30264:
30238:
30184:
30174:
30115:
30089:
30036:
30026:
29952:
29932:
29820:
29810:
29795:
29785:
29744:
29720:
29657:
29647:
29630:
29611:
29578:
29559:
29405:
29385:
29258:
29233:
29087:
29077:
29050:
29040:
29025:
29019:
29003:
28987:
28977:
28950:
28940:
28887:
28881:
28855:
28787:
28777:
28726:
28706:
28650:
28640:
28625:
28615:
28574:
28550:
28479:
28469:
28454:
28444:
28403:
28379:
28311:
28301:
28286:
28276:
28238:
28217:
28154:
28144:
28129:
28119:
28086:
28062:
27994:
27984:
27946:
27930:
27865:
27855:
27819:
27806:
27742:
27732:
27664:
27654:
27634:
27593:
27544:
27494:
27484:
27456:
27446:
27416:
27385:
27344:
27297:
27274:
27260:
27184:
27144:
27106:
27096:
27068:
27058:
27026:
26980:
26957:
26943:
26887:
26841:
26754:
26702:
26656:
26633:
26619:
26570:
26489:
26472:
26451:
26415:
26325:
26308:
26287:
26251:
26115:
26090:
26055:
25923:
25897:
25849:
25759:
25730:
25689:
25575:
25526:
25506:
25475:
25435:{\displaystyle \cos(\mathbf {X} )}
25383:
25349:
25318:
25234:
25203:
25172:
24974:
24936:
24888:
24786:
24755:
24702:
24689:
24679:
24649:
24639:
24629:
24619:
24598:
24557:
24542:
24514:
24460:
24450:
24386:
24358:
24341:
24313:
24266:
24236:
24224:
24108:
24073:
24021:
23973:
23948:
23909:
23891:
23872:
23807:
23768:
23753:
23734:
23717:
23705:
23683:
23641:
23579:
23554:
23537:
23512:
23470:
23366:
23335:
23211:
23189:
23151:
23126:
23063:
23041:
23024:
23002:
22967:
22937:
22846:
22824:
22794:
22772:
22757:
22712:
22701:
22679:
22664:
22604:
22576:
22555:
22502:
22461:
22418:
22407:
22366:
22308:
22272:
22243:
22190:
22155:
22111:
22099:
22087:
22077:
22030:
21995:
21951:
21934:
21924:
21850:
21840:
21788:
21778:
21770:
21754:
21735:
21674:
21664:
21650:
21631:
21577:
21558:
21507:
21499:
21487:
21470:
21458:
21441:
21406:
21380:
21330:
21322:
21310:
21293:
21258:
21241:
21191:
21183:
21163:
21155:
21117:
21106:
21041:
21033:
21016:
21008:
20973:
20953:
20888:
20880:
20842:
20831:
20752:
20713:
20705:
20634:
20626:
20583:
20570:
20560:
20523:
20508:
20480:
20443:
20425:
20415:
20405:
20395:
20375:
20360:
20332:
20285:
20273:
20260:
20250:
20168:
20150:
20140:
20130:
20120:
20055:
20007:
19997:
19987:
19977:
19939:
19883:
19873:
19863:
19853:
19801:
19747:
19736:
19669:
19614:
19552:
19492:
19433:
19375:
19316:
19271:
19209:
19160:
19110:
19049:
19035:
19008:
18850:
18754:
18733:
18702:
18681:
18646:
18627:
18617:
18598:
18588:
18573:
18539:
18529:
18498:
18488:
18455:
18431:
18361:
18340:
18326:
18305:
18293:
18283:
18253:
18243:
18229:
18208:
18194:
18173:
18140:
18108:
18053:
18032:
18020:
18010:
17980:
17970:
17956:
17935:
17902:
17881:
17837:
17823:
17813:
17797:
17770:
17760:
17730:
17720:
17709:
17686:
17672:
17662:
17624:
17606:
17593:
17530:
17520:
17505:
17495:
17462:
17438:
17379:
17365:
17355:
17317:
17310:
17300:
17256:
17240:
17230:
17200:
17190:
17152:
17137:
17078:
17068:
17032:
17019:
16932:
16922:
16847:
16837:
16715:
16658:
16633:
16555:
16508:
16494:
16448:
16401:
16387:
16320:
16276:
16240:
16163:
16139:
16090:
16066:
16040:
15996:
15982:
15961:
15948:
15893:
15883:
15847:
15837:
15803:
15793:
15762:
15752:
15744:
15705:
15693:
15683:
15666:
15642:
15559:
15517:
15492:
15475:
15463:
15453:
15436:
15412:
15350:
15340:
15332:
15315:
15299:
15247:
15203:
15193:
15185:
15168:
15152:
15074:
15032:
15015:
15005:
14944:
14896:
14871:
14832:
14815:
14805:
14759:
14724:
14685:
14668:
14658:
14582:
14543:
14531:
14521:
14504:
14492:
14482:
14447:
14423:
14406:
14382:
14314:
14278:
14268:
14260:
14246:
14207:
14197:
14180:
14170:
14140:
14122:
14112:
14085:
14075:
14041:
14031:
14014:
14004:
13973:
13963:
13955:
13943:
13922:
13912:
13899:
13860:
13843:
13833:
13816:
13787:
13707:
13697:
13680:
13670:
13634:
13624:
13602:
13592:
13558:
13548:
13531:
13521:
13490:
13480:
13472:
13451:
13441:
13433:
13394:
13382:
13372:
13355:
13331:
13266:
13258:
13246:
13229:
13217:
13200:
13165:
13139:
13089:
13081:
13069:
13052:
13017:
13000:
12950:
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12914:
12876:
12865:
12800:
12792:
12775:
12767:
12732:
12712:
12647:
12639:
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12590:
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12471:
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12383:
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12199:
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12084:
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12002:
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11967:
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11880:
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11774:
11764:
11747:
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11702:
11678:
11619:
11601:
11591:
11559:
11549:
11509:
11494:
11437:
11419:
11411:
11394:
11384:
11349:
11341:
11319:
11309:
11271:
11258:
11199:
11181:
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11141:
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11093:
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11017:
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10968:
10955:
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10696:
10623:
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9896:
9875:
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9796:
9781:
9755:
9740:
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9704:
9659:
9644:
9618:
9603:
9582:
9567:
9544:
9529:
9503:
9488:
9467:
9452:
9423:
9413:
9386:
9371:
9352:
9337:
9323:
9308:
9281:
9271:
9237:
9229:
9199:
9191:
9168:
9160:
9131:
9123:
8837:
8819:
8800:
8782:
8768:
8750:
8712:
8694:
8675:
8660:
8646:
8631:
8615:
8597:
8578:
8563:
8549:
8534:
8507:
8497:
8433:
8425:
8396:
8388:
8364:
8356:
8308:
8300:
8274:
8266:
8245:
8237:
8211:
8203:
8177:
8169:
8148:
8140:
8111:
8103:
8069:
8054:
8028:
8013:
7992:
7977:
7932:
7917:
7891:
7876:
7855:
7840:
7817:
7802:
7776:
7761:
7740:
7725:
7696:
7686:
7659:
7644:
7621:
7606:
7590:
7575:
7548:
7538:
7504:
7496:
7470:
7462:
7441:
7433:
7404:
7396:
7324:
7314:
7280:
7270:
7226:
7218:
7133:
7123:
7079:
7069:
7028:
7020:
6937:
6927:
6888:
6878:
6845:
6837:
6691:
6681:
6650:
6642:
6625:
6615:
6600:
6590:
6573:
6565:
6522:
6514:
6474:
6466:
6431:
6421:
6378:
6370:
6335:
6325:
6280:
6270:
6214:
6204:
6173:
6165:
6129:
6119:
6082:
6072:
6036:
6028:
5996:
5986:
5955:
5947:
5910:
5900:
5868:
5858:
5827:
5819:
5782:
5772:
5762:If we choose numerator layout for
5736:
5726:
5695:
5687:
5635:
5625:
5585:
5575:
5537:
5529:
5452:
5442:
5389:
5379:
5243:
5233:
5153:
5145:
5114:
5048:
5029:
4998:
4929:
4921:
4892:
4884:
4860:
4852:
4804:
4796:
4770:
4762:
4741:
4733:
4707:
4699:
4673:
4665:
4644:
4636:
4611:
4603:
4526:
4508:
4489:
4471:
4457:
4439:
4401:
4383:
4364:
4349:
4335:
4320:
4304:
4286:
4267:
4252:
4238:
4223:
4200:
4190:
4087:
4077:
3956:
3941:
3915:
3900:
3879:
3864:
3819:
3804:
3778:
3763:
3742:
3727:
3704:
3689:
3663:
3648:
3627:
3612:
3587:
3577:
3333:
3325:
3305:
3259:
3229:
3140:
3132:
3097:
3089:
3059:
3051:
3031:
2952:
2944:
2916:
2908:
2885:
2877:
2851:
2843:
2657:
2647:
1929:
1921:
1882:
1872:
1840:
1832:
1797:
1787:
1757:
1747:
1717:
1709:
1599:
1591:
1568:
1560:
1537:
1529:
1490:
1482:
1465:
1283:
1275:
1234:
1226:
1185:
1177:
1165:
44:Part of a series of articles about
25:
36867:
36647:The Method of Mechanical Theorems
35650:
35410:Generalized multivariate analysis
35311:
31686:{\displaystyle d(a\mathbf {X} )=}
31561:Differential identities: matrix
29497:Derivative of the exponential map
26375:{\displaystyle 2\mathbf {X} ^{+}}
25809:{\displaystyle \mathbf {X} ^{-1}}
6194:and matrix-by-scalar derivatives
1085:As a first example, consider the
36202:Partial fractions in integration
36118:Stochastic differential equation
35734:(slide presentation), Zhang Le,
35629:Journal of Multivariate Analysis
35518:Journal of Multivariate Analysis
35283:
35227:
34639:
34587:
34525:
34494:multivariate normal distribution
34462:
34443:
34407:
34399:
34371:
34360:
34350:
34321:
34312:
34304:
34276:
34257:
34221:
34213:
34185:
34174:
34134:
34125:
34076:
34064:
34027:
34013:
33868:
33840:
33828:
33798:
33769:
33741:
33700:
33637:
33583:
33549:
33543:
33538:
33530:
33466:
33444:
33414:
33176:
33167:
33150:
33102:
33021:
32993:
32981:
32947:
32918:
32873:
32861:
32850:
32826:
32814:
32715:
32687:
32674:
32619:
32562:
32553:
32536:
32472:
32421:
32377:
32328:
32317:
32295:
32252:
32207:
32165:
32127:
32113:
32105:
32094:
32059:
32051:
32010:
31996:
31988:
31977:
31942:
31934:
31893:
31882:
31874:
31866:
31831:
31826:
31793:
31782:
31750:
31742:
31716:{\displaystyle a\,d\mathbf {X} }
31709:
31673:
31614:{\displaystyle d(\mathbf {A} )=}
31601:
31532:
31515:
31469:
31424:
31413:
31382:
31365:
31343:
31304:
31265:
31221:
31141:
31128:
31074:
31004:
30989:
30983:
30929:
30918:
30768:
30744:
30731:
30664:
30650:
30637:
30574:
30490:
30470:
30429:
30409:
30359:
30332:
30306:
30253:
30178:
30158:
30104:
30030:
30010:
29988:
29941:
29833:
29814:
29789:
29775:
29735:
29727:
29651:
29634:
29621:
29569:
29474:
29467:
29449:
29436:
29397:
29345:
29337:
29322:
29310:
29295:
29289:
29249:
29238:
29103:
29081:
29061:
29044:
29013:
28981:
28961:
28944:
28919:
28866:
28798:
28781:
28761:
28711:
28663:
28644:
28619:
28605:
28565:
28557:
28492:
28473:
28448:
28434:
28394:
28386:
28321:
28305:
28280:
28269:
28229:
28224:
28148:
28123:
28077:
28069:
28004:
27988:
27977:
27940:
27937:
27934:
27859:
27813:
27736:
27669:
27660:
27650:
27639:
27630:
27601:
27598:
27589:
27578:
27575:
27556:
27551:
27540:
27490:
27480:
27461:
27452:
27442:
27421:
27412:
27393:
27390:
27381:
27356:
27351:
27340:
27301:
27286:
27281:
27270:
27192:
27189:
27180:
27169:
27166:
27156:
27151:
27140:
27102:
27092:
27073:
27064:
27054:
27038:
27033:
27022:
26984:
26969:
26964:
26953:
26869:
26853:
26848:
26837:
26796:
26785:
26770:
26750:
26725:
26714:
26709:
26698:
26660:
26645:
26640:
26629:
26565:
26535:
26494:
26478:
26467:
26400:
26362:
26329:
26314:
26303:
26233:
26212:
26172:
26156:
26119:
26099:
26037:
26021:
26018:
26015:
25978:
25967:
25964:
25961:
25927:
25912:
25909:
25906:
25831:
25793:
25763:
25748:
25671:
25655:
25600:
25589:
25530:
25515:
25463:
25425:
25387:
25371:
25307:
25281:{\displaystyle e^{\mathbf {X} }}
25272:
25238:
25222:
25144:
25138:
25127:
25052:
25046:
25035:
24978:
24957:
24951:
24867:
24823:
24790:
24770:
24721:
24718:
24715:
24712:
24698:
24694:
24685:
24675:
24645:
24635:
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24615:
24606:
24603:
24594:
24590:
24561:
24547:
24538:
24534:
24531:
24528:
24456:
24446:
24422:
24419:
24390:
24377:
24374:
24371:
24345:
24332:
24329:
24326:
24248:
24231:
24206:
24165:
24159:
24145:
24112:
24099:
24094:
24091:
24087:
24032:
24016:
24007:
23968:
23959:
23943:
23913:
23899:
23896:
23887:
23831:
23802:
23772:
23758:
23749:
23721:
23701:
23697:
23636:
23612:
23583:
23570:
23567:
23541:
23528:
23525:
23457:
23445:
23412:
23400:
23370:
23354:
23348:
23215:
23202:
23155:
23142:
23067:
23054:
23028:
23015:
22971:
22958:
22950:
22879:
22850:
22837:
22789:
22783:
22778:
22767:
22752:
22746:
22741:
22736:
22696:
22690:
22685:
22674:
22659:
22653:
22648:
22643:
22608:
22595:
22590:
22582:
22567:
22562:
22497:
22488:
22480:
22475:
22456:
22447:
22402:
22393:
22385:
22380:
22361:
22352:
22312:
22299:
22291:
22286:
22278:
22263:
22255:
22250:
22185:
22179:
22150:
22144:
22115:
22105:
22094:
22082:
22025:
22019:
21990:
21984:
21955:
21945:
21940:
21929:
21844:
21782:
21758:
21745:
21668:
21654:
21641:
21568:
21511:
21410:
21334:
21262:
21195:
21167:
21121:
21045:
21020:
20977:
20892:
20846:
20776:
20747:
20717:
20638:
20579:
20575:
20566:
20556:
20547:
20544:
20541:
20538:
20527:
20513:
20504:
20500:
20497:
20494:
20451:
20448:
20439:
20435:
20421:
20411:
20401:
20391:
20379:
20365:
20356:
20352:
20349:
20346:
20294:
20269:
20265:
20256:
20246:
20237:
20234:
20231:
20228:
20191:
20178:
20175:
20164:
20160:
20146:
20136:
20126:
20116:
20076:
20065:
20062:
20051:
20047:
20020:
20003:
19993:
19983:
19973:
19935:
19931:
19923:
19912:
19909:
19879:
19869:
19859:
19849:
19840:
19797:
19793:
19785:
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19771:
19727:
19716:
19713:
19710:
19707:
19665:
19661:
19653:
19640:
19637:
19605:
19594:
19591:
19588:
19585:
19548:
19544:
19536:
19525:
19522:
19488:
19475:
19472:
19469:
19466:
19429:
19425:
19417:
19414:
19411:
19371:
19367:
19355:
19352:
19349:
19312:
19308:
19300:
19297:
19294:
19267:
19263:
19252:
19249:
19246:
19205:
19201:
19198:
19195:
19192:
19156:
19152:
19149:
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19143:
19115:
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19099:
19096:
19053:
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19024:
19021:
18974:
18971:
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18916:
18913:
18910:
18907:
18883:
18880:
18877:
18874:
18846:
18820:
18758:
18745:
18737:
18706:
18693:
18685:
18641:
18621:
18592:
18568:
18533:
18519:
18511:
18492:
18446:
18438:
18365:
18352:
18344:
18330:
18317:
18309:
18287:
18247:
18233:
18220:
18212:
18198:
18185:
18177:
18128:
18120:
18112:
18057:
18044:
18036:
18014:
17974:
17960:
17947:
17939:
17893:
17885:
17817:
17792:
17783:
17764:
17724:
17704:
17695:
17666:
17615:
17601:
17524:
17499:
17453:
17445:
17359:
17305:
17251:
17234:
17194:
17183:
17146:
17141:
17072:
17026:
16963:
16926:
16841:
16792:
16784:
16774:
16766:
16734:
16726:
16706:
16698:
16663:
16649:
16641:
15897:
15887:
15859:
15851:
15841:
15807:
15797:
15766:
15756:
15739:
15709:
15699:
15688:
15670:
15657:
15649:
15586:
15554:
15521:
15501:
15479:
15469:
15458:
15440:
15427:
15419:
15383:
15345:
15336:
15326:
15321:
15310:
15242:
15233:
15198:
15189:
15179:
15174:
15163:
15111:
15106:
15080:
15069:
15036:
15026:
15021:
15010:
14955:
14939:
14930:
14891:
14882:
14866:
14836:
14826:
14821:
14810:
14765:
14754:
14730:
14719:
14689:
14679:
14674:
14663:
14606:
14577:
14547:
14537:
14526:
14508:
14498:
14487:
14451:
14438:
14430:
14410:
14397:
14389:
14338:
14309:
14273:
14264:
14211:
14201:
14184:
14174:
14146:
14135:
14126:
14116:
14102:
14097:
14089:
14079:
14045:
14035:
14018:
14008:
13977:
13967:
13950:
13938:
13926:
13916:
13905:
13894:
13864:
13854:
13849:
13838:
13820:
13807:
13802:
13794:
13711:
13701:
13684:
13674:
13646:
13638:
13628:
13614:
13606:
13596:
13562:
13552:
13535:
13525:
13494:
13484:
13467:
13455:
13445:
13428:
13398:
13388:
13377:
13359:
13346:
13338:
13270:
13169:
13093:
13021:
12954:
12926:
12880:
12804:
12779:
12736:
12651:
12605:
12534:
12505:
12475:
12388:
12372:
12315:
12302:
12294:
12280:
12267:
12259:
12245:
12235:
12203:
12193:
12179:
12166:
12158:
12144:
12131:
12123:
12088:
12072:
12064:
12056:
12006:
11993:
11985:
11971:
11961:
11929:
11919:
11905:
11892:
11884:
11849:
11836:
11828:
11778:
11768:
11751:
11741:
11706:
11693:
11685:
11614:
11605:
11595:
11563:
11553:
11542:
11513:
11503:
11498:
11432:
11423:
11398:
11388:
11353:
11334:
11323:
11313:
11275:
11265:
11194:
11185:
11145:
11126:
11097:
11087:
11021:
11011:
10973:
10962:
10899:
10870:
10840:
10830:
10819:
10768:
10744:
10715:
10705:
10700:
10656:
10627:
10617:
10584:
10555:
10545:
10462:
10452:
9842:
9427:
9417:
9275:
9135:
8865:
8501:
8115:
7700:
7690:
7542:
7408:
7328:
7318:
7284:
7274:
7230:
7137:
7127:
7083:
7073:
7032:
6931:
6882:
6729:multivariate normal distribution
6685:
6654:
6629:
6619:
6594:
6577:
6527:
6478:
6425:
6382:
6329:
6274:
6208:
6177:
6133:
6123:
6109:the vector-by-vector derivative
6076:
6041:
5990:
5959:
5914:
5904:
5862:
5831:
5786:
5776:
5730:
5699:
5640:
5629:
5589:
5579:
5541:
5456:
5446:
5418:
5393:
5383:
5247:
5237:
5165:
5157:
5119:
5052:
5039:
5016:
5003:
4615:
4194:
4102:
4091:
4081:
4064:
4056:
3591:
3581:
3479:
3390:
3345:
3337:
3310:
3280:
3269:
3249:
3234:
3144:
2855:
2745:
2651:
2463:
2352:
1953:
1951:
1933:
1904:
1886:
1876:
1844:
1791:
1751:
1494:
1441:
1370:represents a unit vector in the
36340:Jacobian matrix and determinant
36195:Tangent half-angle substitution
36163:Fundamental theorem of calculus
35719:North Carolina State University
35538:
35505:
35480:
35455:
35426:
35398:
35386:
34483:
33686:for diagonalizable matrices by
33647:{\displaystyle f(\mathbf {X} )}
31170:Identities in differential form
29529:ignores row vs. column layout)
2239:
36416:Arithmetico-geometric sequence
36108:Ordinary differential equation
35692:
35344:
35331:
35305:
35101:
34942:
34918:
34743:
34734:
34662:
34138:
34121:
33736:
33723:
33704:
33696:
33673:
33667:
33641:
33633:
33588:
33579:
33480:
33461:
33449:
33440:
33425:
33409:
33288:
33275:
33266:
33253:
33215:
33202:
33171:
33160:
33106:
33098:
32878:
32857:
32854:
32843:
32831:
32810:
32711:
32699:
32691:
32680:
32557:
32546:
32426:
32414:
32212:
32200:
32131:
32120:
32098:
32087:
32063:
32047:
32014:
32003:
31981:
31970:
31946:
31930:
31897:
31886:
31870:
31859:
31835:
31822:
31754:
31738:
31677:
31666:
31605:
31597:
31478:
31474:
31464:
31454:
31428:
31417:
31409:
31400:
31348:
31338:
31313:
31309:
31299:
31295:
31269:
31258:
31228:
31225:
31217:
31208:
31008:
30997:
30936:
30933:
30922:
30914:
30735:
30727:
30641:
30633:
30578:
30570:
30258:
30248:
30109:
30099:
29993:
29983:
29946:
29936:
29739:
29723:
29625:
29617:
29573:
29565:
29341:
29330:
29314:
29303:
29253:
29242:
28569:
28553:
28398:
28382:
28233:
28220:
28081:
28065:
27291:
27264:
26790:
26780:
26775:
26765:
26542:{\displaystyle -2\mathbf {X} }
26026:
26010:
25972:
25956:
25917:
25901:
25753:
25740:
25660:
25650:
25643:
25637:
25594:
25584:
25571:
25564:
25520:
25510:
25471:
25467:
25459:
25450:
25429:
25421:
25378:
25375:
25367:
25358:
24381:
24367:
24336:
24322:
24103:
24082:
23574:
23563:
23532:
23521:
23461:
23453:
23416:
23408:
23361:
23344:
23206:
23198:
23146:
23135:
23058:
23050:
23019:
23011:
22962:
22946:
22841:
22833:
22599:
22586:
22572:
22558:
22492:
22471:
22397:
22376:
22303:
22282:
22268:
22246:
21749:
21741:
21645:
21637:
21572:
21564:
21482:
21476:
21453:
21447:
21401:
21398:
21392:
21386:
21305:
21299:
21253:
21247:
20968:
20956:
20080:
20069:
20024:
20013:
19927:
19916:
19844:
19833:
19789:
19778:
19732:
19720:
19657:
19633:
19610:
19598:
19540:
19529:
19421:
19407:
19304:
19259:
19119:
19092:
19044:
19017:
18978:
18961:
18949:
18932:
18920:
18903:
18887:
18870:
18824:
18816:
18749:
18741:
18697:
18689:
18450:
18434:
18356:
18348:
18321:
18313:
18224:
18216:
18189:
18181:
18135:
18132:
18124:
18116:
18048:
18040:
17951:
17943:
17897:
17889:
17619:
17596:
17457:
17441:
16711:
16694:
16594:
16567:
16540:
16513:
16444:
16416:
16383:
16355:
16315:
16288:
16272:
16244:
15661:
15645:
15505:
15497:
15431:
15415:
14442:
14426:
14401:
14385:
13811:
13790:
13350:
13334:
13241:
13235:
13212:
13206:
13160:
13157:
13151:
13145:
13064:
13058:
13012:
13006:
12727:
12715:
12306:
12298:
12271:
12263:
12170:
12162:
12135:
12127:
12079:
12076:
12068:
12060:
11997:
11989:
11896:
11888:
11840:
11832:
11697:
11681:
5043:
5035:
5020:
5012:
4068:
4060:
3273:
3265:
3253:
3245:
1363:{\displaystyle {\hat {x}}_{i}}
1348:
1306:
1257:
1208:
1142:
1103:
1049:with respect to a vector as a
129:
123:
114:
108:
92:
86:
13:
1:
36239:Integro-differential equation
36113:Partial differential equation
35339:Determinant § Derivative
35062:
34507:to compute, for example, the
27722:Identities: matrix-by-scalar
24429:{\displaystyle \mathbf {BA} }
20616:Identities: scalar-by-matrix
16827:Identities: vector-by-scalar
15593:{\displaystyle 2\mathbf {x} }
15390:{\displaystyle 2\mathbf {A} }
12350:Identities: scalar-by-vector
10438:Identities: vector-by-vector
10207:
5337:, i.e. lay out according to
4015:pushforward (or differential)
1976:Relation to other derivatives
1422:{\displaystyle 1\leq i\leq 3}
1051:column vector or a row vector
429:Integral of inverse functions
35799:
35725:Matrix Differential Calculus
34646:{\displaystyle \mathbf {0} }
34594:{\displaystyle \mathbf {0} }
34541:Derivative (generalizations)
33391:{\displaystyle \delta _{ij}}
33071:{\displaystyle \lambda _{i}}
29873:Consistent numerator layout,
23838:{\displaystyle \mathbf {A} }
23619:{\displaystyle \mathbf {A} }
22886:{\displaystyle \mathbf {I} }
20783:{\displaystyle \mathbf {0} }
20673:Denominator layout, i.e. by
20606:(For the last step, see the
16970:{\displaystyle \mathbf {0} }
16887:Denominator layout, i.e. by
14613:{\displaystyle \mathbf {a} }
14345:{\displaystyle \mathbf {H} }
12541:{\displaystyle \mathbf {0} }
10906:{\displaystyle \mathbf {A} }
10751:{\displaystyle \mathbf {A} }
10663:{\displaystyle \mathbf {I} }
10591:{\displaystyle \mathbf {0} }
10503:Denominator layout, i.e. by
5485:, i.e. lay out according to
5425:{\displaystyle \mathbf {y} }
3176:By example, in physics, the
1448:{\displaystyle \mathbf {x} }
1433:, with respect to a vector,
998:, especially over spaces of
7:
36393:Generalized Stokes' theorem
36180:Integration by substitution
35655:
35362:10.1007/978-3-540-68942-3_4
34518:
33905:Canonical differential form
29502:Scalar-by-scalar identities
27715:Matrix-by-scalar identities
18779:Scalar-by-matrix identities
16820:Vector-by-scalar identities
12340:Scalar-by-vector identities
10427:Vector-by-vector identities
9102:Denominator-layout notation
6313:Consistent numerator layout
5097:in the direction of matrix
4143:numerator layout convention
2613:the derivative of a vector
2323:numerator layout convention
2220:numerator layout convention
2030:
1678:Types of matrix derivative
1002:. It collects the various
847:Calculus on Euclidean space
270:Logarithmic differentiation
10:
36872:
35922:(ε, δ)-definition of limit
35641:10.1016/j.jmva.2010.05.005
35555:10.1007/s00362-023-01499-w
35531:10.1016/j.jmva.2021.104849
35441:. Beijing: Science Press.
35408:; Zhang, Yao-Ting (1990).
34490:multivariate distributions
31572:Result (numerator layout)
31194:Result (numerator layout)
29494:
27771:Numerator layout, i.e. by
20665:Numerator layout, i.e. by
16876:Numerator layout, i.e. by
12436:; result is column vector
10489:Numerator layout, i.e. by
5359:. This corresponds to the
2264:
29:
36815:Proof that 22/7 exceeds π
36752:
36730:
36656:
36604:Gottfried Wilhelm Leibniz
36574:
36551:e (mathematical constant)
36536:
36408:
36315:
36247:
36128:
35930:
35885:
35807:
34511:for the case of multiple
31109:
30964:
30294:
30138:
29975:
26441:
26277:
25887:
23176:
22992:
22871:
21825:
21547:
21531:
21431:
21283:
21142:
20998:
20867:
18385:
18098:
18082:
18077:
17871:
17855:
17485:
17340:
17055:
16955:
15372:
15225:
13190:
13042:
12901:
12757:
12626:
11727:
10994:
10648:
10576:
7375:Numerator-layout notation
7346:
7304:
7302:
7260:
7208:
7185:
7155:
7113:
7059:
7010:
6987:
6917:
6868:
6865:
6827:
6816:
6775:
6759:
6751:
6104:and use numerator layout.
5432:and the column number of
5318:laid out in rows and the
4588:numerator layout notation
4175:numerator layout notation
4122:Derivatives with matrices
4006:with respect to a vector
3562:numerator layout notation
2828:numerator layout notation
2619:with respect to a scalar
2444:numerator layout notation
2278:by identifying the space
1996:
1908:
1819:
1696:
1691:
1688:
1685:
1682:
581:Summand limit (term test)
36566:Stirling's approximation
36039:Implicit differentiation
35987:Rules of differentiation
35715:Notes on Matrix Calculus
35682:matrix expression module
35491:. New York: John Wiley.
34561:
34498:elliptical distributions
33941:{\displaystyle dy=a\,dx}
26607:is square and invertible
26080:
25496:
25339:
25193:
24926:
24745:
24063:
23862:
23673:
23502:
21548:
18998:For example, to compute
16884:result is column vector
10406:are functions of one of
10345:are functions of one of
10284:are functions of one of
5976:as a column vector, and
5509:, in distinction to the
5206:Other matrix derivatives
2261:Derivatives with vectors
1072:
265:Implicit differentiation
255:Differentiation notation
182:Inverse function theorem
30:Not to be confused with
36800:Euler–Maclaurin formula
36705:trigonometric functions
36158:Constant of integration
35736:University of Edinburgh
35703:Imperial College London
35699:Matrix Reference Manual
35686:array expression module
35466:. Dordrecht: Springer.
30879:is its derivative, and
29207:is its derivative, and
23309:is its derivative, and
12425:; result is row vector
4033:with respect to vector
3180:is the negative vector
2201:differentiability class
723:Helmholtz decomposition
36856:Multivariable calculus
36769:Differential geometry
36614:Infinitesimal calculus
36317:Multivariable calculus
36265:Directional derivative
36071:Second derivative test
36049:Logarithmic derivative
36022:General Leibniz's rule
35917:Order of approximation
35758:Vector/Matrix Calculus
35748:), Heino Bohn Nielsen.
35297:
35249:
35050:
34867:
34647:
34595:
34470:
34422:
34379:
34329:
34284:
34236:
34193:
34145:
34090:
34035:
33986:
33942:
33882:
33812:
33755:
33680:
33648:
33617:
33566:
33496:
33392:
33355:
33121:
33072:
33035:
32961:
32901:
32781:
32732:
32636:
32588:
32533:
32493:
32441:
32400:
32345:
32276:
32225:
32186:
32138:
32073:
32021:
31956:
31904:
31845:
31801:
31764:
31717:
31687:
31636:
31615:
31545:
31488:
31435:
31323:
31276:
31238:
31156:
31102:
31020:
30957:
30792:
30688:
30599:
30529:
30287:
30202:
30131:
30054:
29968:
29855:With matrices involved
29841:
29760:
29670:
29594:
29482:
29421:
29353:
29274:
29120:
28903:
28815:
28742:
28671:
28590:
28500:
28419:
28329:
28254:
28167:
28102:
28012:
27962:
27878:
27835:
27755:
27701:
27515:
27315:
27213:
27116:
26998:
26896:
26813:
26674:
26579:
26543:
26514:
26424:
26376:
26343:
26260:
26189:
26133:
26064:
25995:
25941:
25858:
25810:
25777:
25698:
25617:
25544:
25484:
25436:
25401:
25327:
25282:
25252:
25181:
25118:
25078:
25032:
24992:
24897:
24843:
24804:
24729:
24659:
24575:
24470:
24430:
24404:
24275:
24182:
24126:
24040:
23987:
23927:
23839:
23816:
23786:
23650:
23620:
23597:
23479:
23423:
23384:
23226:
23169:
23078:
22985:
22887:
22864:
22803:
22721:
22622:
22511:
22427:
22326:
22199:
22164:
22129:
22039:
22004:
21969:
21876:
21816:
21702:
21603:
21522:
21424:
21345:
21276:
21206:
21135:
21056:
20991:
20903:
20860:
20784:
20761:
20731:
20649:
20596:
20462:
20311:
19067:
18989:
18769:
18717:
18655:
18552:
18471:
18376:
18266:
18156:
18068:
17993:
17918:
17846:
17743:
17640:
17543:
17478:
17388:
17333:
17265:
17213:
17168:
17091:
17048:
16971:
16948:
16860:
16806:
16748:
16677:
16601:
16471:
16341:
16184:
16104:
16017:
15914:in denominator layout
15908:
15867:
15818:
15777:
15723:
15594:
15568:
15535:
15391:
15365:
15256:
15218:
15119:
15088:
15050:
14963:
14910:
14850:
14773:
14738:
14703:
14614:
14591:
14561:
14465:
14346:
14323:
14293:
14228:in denominator layout
14222:
14154:
14056:
13988:
13878:
13728:in denominator layout
13722:
13654:
13573:
13505:
13412:
13281:
13183:
13104:
13035:
12965:
12894:
12815:
12750:
12662:
12619:
12542:
12519:
12489:
12401:
12326:
12214:
12102:
12017:
11940:
11863:
11789:
11720:
11628:
11574:
11527:
11446:
11364:
11289:
11208:
11156:
11111:
11032:
10987:
10907:
10884:
10854:
10782:
10752:
10729:
10664:
10641:
10592:
10569:
10473:
10198:
9092:
8471:
7339:
7295:
7241:
7148:
7094:
7043:
6950:
6901:
6858:
6704:
6665:
6545:
6489:
6444:
6393:
6348:
6296:
6230:
6188:
6147:
6098:
6056:
6009:
5970:
5934:we should lay out the
5928:
5881:
5842:
5806:we should lay out the
5800:
5752:
5716:and the opposite case
5710:
5658:
5606:equals to the size of
5600:
5555:
5473:equals to the size of
5467:
5426:
5404:
5258:
5194:, particularly in the
5181:
5078:directional derivative
5063:
4963:
4550:
4113:
3987:
3554:
3465:
3356:
3288:
3193:directional derivative
3167:
2984:
2820:
2670:
2600:
2427:
2097:, etc. An element of
1944:
1897:
1855:
1810:
1770:
1730:
1639:
1449:
1423:
1391:
1364:
1325:
1149:
1024:differential equations
996:multivariable calculus
857:Limit of distributions
677:Directional derivative
338:Faà di Bruno's formula
136:
36688:logarithmic functions
36683:exponential functions
36599:Generality of algebra
36477:Tests of convergence
36103:Differential equation
36087:Further applications
36076:Extreme value theorem
36066:First derivative test
35960:Differential operator
35932:Differential calculus
35319:. Stanford University
35298:
35250:
35051:
34868:
34648:
34596:
34513:explanatory variables
34471:
34423:
34380:
34330:
34285:
34237:
34194:
34146:
34091:
34036:
33987:
33943:
33883:
33813:
33756:
33681:
33649:
33618:
33567:
33497:
33393:
33356:
33122:
33073:
33036:
32962:
32902:
32782:
32733:
32637:
32589:
32507:
32494:
32454:is a positive integer
32442:
32401:
32346:
32277:
32226:
32187:
32139:
32074:
32022:
31957:
31905:
31846:
31802:
31765:
31718:
31688:
31649:is not a function of
31637:
31616:
31580:is not a function of
31546:
31489:
31436:
31324:
31277:
31239:
31157:
31103:
31035:is not a function of
31021:
30958:
30807:is not a function of
30793:
30689:
30600:
30530:
30288:
30203:
30132:
30055:
29969:
29842:
29761:
29671:
29595:
29507:With vectors involved
29483:
29422:
29368:is not a function of
29354:
29275:
29135:is not a function of
29121:
28904:
28816:
28743:
28672:
28591:
28501:
28420:
28330:
28255:
28168:
28103:
28013:
27963:
27899:are not functions of
27879:
27836:
27756:
27702:
27516:
27316:
27228:is not a function of
27214:
27117:
26999:
26911:is not a function of
26897:
26814:
26675:
26594:is not a function of
26580:
26544:
26515:
26425:
26377:
26344:
26261:
26190:
26134:
26077:is a positive integer
26065:
25996:
25942:
25879:are not functions of
25859:
25811:
25778:
25713:is not a function of
25699:
25618:
25545:
25485:
25437:
25402:
25328:
25283:
25253:
25182:
25092:
25079:
25006:
24993:
24923:is a positive integer
24912:is not a function of
24898:
24844:
24805:
24742:is a positive integer
24730:
24660:
24576:
24497:are not functions of
24471:
24431:
24405:
24296:are not functions of
24276:
24183:
24127:
24055:is not a function of
24041:
23988:
23928:
23854:is not a function of
23840:
23817:
23787:
23665:is not a function of
23651:
23621:
23598:
23494:is not a function of
23480:
23424:
23385:
23227:
23170:
23093:is not a function of
23079:
22986:
22888:
22865:
22804:
22722:
22623:
22538:are not functions of
22512:
22428:
22327:
22226:are not functions of
22200:
22165:
22130:
22060:are not functions of
22040:
22005:
21970:
21907:are not functions of
21877:
21817:
21703:
21604:
21523:
21425:
21346:
21277:
21207:
21136:
21057:
20992:
20904:
20861:
20799:is not a function of
20785:
20762:
20732:
20688:is not a function of
20650:
20597:
20463:
20312:
19068:
18990:
18770:
18718:
18656:
18553:
18472:
18377:
18267:
18157:
18069:
17994:
17919:
17847:
17744:
17641:
17544:
17479:
17389:
17334:
17266:
17214:
17169:
17106:is not a function of
17092:
17049:
16986:is not a function of
16972:
16949:
16905:is not a function of
16895:result is row vector
16861:
16807:
16749:
16678:
16616:is not a function of
16602:
16472:
16342:
16223:are not functions of
16185:
16105:
16018:
15931:are not functions of
15909:
15868:
15819:
15778:
15724:
15609:is not a function of
15595:
15569:
15536:
15392:
15366:
15271:is not a function of
15257:
15219:
15134:is not a function of
15120:
15089:
15051:
14978:is not a function of
14964:
14911:
14851:
14788:is not a function of
14774:
14739:
14704:
14641:is not a function of
14629:is not a function of
14615:
14592:
14562:
14466:
14365:is not a function of
14347:
14324:
14294:
14223:
14155:
14057:
13989:
13879:
13770:is not a function of
13723:
13655:
13574:
13506:
13413:
13282:
13184:
13105:
13036:
12966:
12895:
12816:
12751:
12663:
12620:
12557:is not a function of
12543:
12520:
12490:
12446:is not a function of
12402:
12327:
12215:
12103:
12018:
11941:
11864:
11790:
11721:
11629:
11575:
11528:
11461:is not a function of
11447:
11365:
11290:
11209:
11157:
11112:
11063:is not a function of
11033:
10988:
10922:is not a function of
10908:
10885:
10855:
10797:is not a function of
10783:
10753:
10730:
10679:is not a function of
10665:
10642:
10593:
10570:
10524:is not a function of
10474:
10199:
9093:
8472:
7340:
7296:
7242:
7149:
7095:
7044:
6951:
6902:
6859:
6705:
6666:
6546:
6490:
6445:
6394:
6349:
6297:
6231:
6189:
6148:
6099:
6057:
6010:
5971:
5929:
5882:
5848:as a row vector, and
5843:
5801:
5753:
5711:
5659:
5601:
5556:
5468:
5427:
5405:
5259:
5182:
5064:
4964:
4551:
4114:
3988:
3555:
3466:
3357:
3289:
3195:of a scalar function
3168:
2985:
2821:
2671:
2601:
2428:
2246:tensor index notation
2130:, etc. An element of
1945:
1898:
1856:
1811:
1771:
1731:
1640:
1450:
1424:
1392:
1390:{\displaystyle x_{i}}
1365:
1326:
1150:
1036:tensor index notation
1016:multivariate function
1010:with respect to many
941:Mathematical analysis
852:Generalized functions
537:arithmetico-geometric
383:Leibniz integral rule
137:
36753:Miscellaneous topics
36693:hyperbolic functions
36678:irrational functions
36556:Exponential function
36409:Sequences and series
36175:Integration by parts
35265:
35217:
34877:
34678:
34635:
34605:of all 0's, of size
34583:
34433:
34392:
34340:
34297:
34247:
34206:
34156:
34103:
34046:
33999:
33953:
33916:
33823:
33765:
33690:
33679:{\displaystyle f(x)}
33661:
33627:
33576:
33526:
33406:
33372:
33132:
33084:
33055:
33051:at every eigenvalue
32976:
32914:
32792:
32747:
32647:
32603:
32504:
32460:
32411:
32365:
32287:
32240:
32197:
32153:
32084:
32041:
31967:
31924:
31856:
31816:
31775:
31732:
31698:
31660:
31626:
31591:
31499:
31448:
31334:
31289:
31249:
31202:
31113:
31046:
30968:
30899:
30699:
30610:
30558:
30298:
30231:
30142:
30083:
29979:
29926:
29771:
29714:
29605:
29553:
29525:Any layout (assumes
29432:
29379:
29285:
29227:
28914:
28848:
28753:
28700:
28601:
28544:
28430:
28373:
28265:
28211:
28113:
28056:
27973:
27924:
27846:
27800:
27726:
27526:
27326:
27254:
27127:
27009:
26937:
26824:
26685:
26613:
26554:
26525:
26445:
26387:
26354:
26281:
26200:
26144:
26084:
26006:
25952:
25891:
25821:
25788:
25724:
25628:
25555:
25500:
25447:
25412:
25343:
25293:
25263:
25197:
25089:
25003:
24930:
24854:
24815:
24749:
24670:
24586:
24508:
24441:
24415:
24307:
24193:
24137:
24067:
23998:
23938:
23866:
23827:
23797:
23677:
23631:
23608:
23506:
23434:
23395:
23329:
23180:
23120:
22996:
22931:
22875:
22818:
22732:
22633:
22549:
22438:
22337:
22237:
22175:
22140:
22071:
22015:
21980:
21918:
21834:
21713:
21614:
21552:
21435:
21374:
21287:
21235:
21146:
21100:
21002:
20947:
20871:
20825:
20772:
20742:
20699:
20620:
20602:(denominator layout)
20474:
20326:
19076:
19002:
18803:
18727:
18675:
18563:
18482:
18425:
18277:
18167:
18102:
18004:
17929:
17875:
17754:
17651:
17587:
17489:
17432:
17344:
17294:
17224:
17179:
17131:
17059:
17013:
16959:
16916:
16831:
16759:
16688:
16627:
16482:
16352:
16234:
16115:
16028:
15942:
15877:
15831:
15824:in numerator layout
15787:
15734:
15636:
15579:
15546:
15406:
15376:
15292:
15229:
15145:
15099:
15061:
14999:
14921:
14861:
14799:
14749:
14714:
14652:
14602:
14572:
14476:
14376:
14334:
14304:
14239:
14164:
14069:
14062:in numerator layout
13998:
13889:
13781:
13664:
13586:
13579:in numerator layout
13515:
13423:
13325:
13194:
13133:
13046:
12994:
12905:
12859:
12761:
12706:
12630:
12584:
12530:
12500:
12457:
12354:
12225:
12113:
12046:
11951:
11874:
11818:
11731:
11672:
11585:
11538:
11488:
11375:
11300:
11252:
11167:
11122:
11074:
10998:
10949:
10895:
10865:
10808:
10763:
10740:
10690:
10652:
10607:
10580:
10535:
10442:
9113:
8487:
7386:
7308:
7264:
7212:
7117:
7063:
7014:
6921:
6872:
6831:
6675:
6559:
6508:
6460:
6415:
6364:
6319:
6264:
6198:
6159:
6113:
6066:
6022:
5980:
5941:
5894:
5852:
5813:
5766:
5720:
5681:
5619:
5569:
5523:
5497:(i.e. contrarily to
5436:
5414:
5373:
5357:Jacobian formulation
5349:(i.e. contrarily to
5276:and the denominator
5227:
5110:
5072:Also in analog with
4994:
4976:of a matrix and the
4597:
4570:, with respect to a
4184:
4049:
3571:
3475:
3386:
3378:The derivative of a
3301:
3225:
3206:of the space vector
3028:
2837:
2741:
2641:
2453:
2348:
2054:denote the space of
2003:Lagrange multipliers
1915:
1866:
1826:
1781:
1741:
1703:
1462:
1437:
1401:
1374:
1338:
1162:
1097:
946:Nonstandard analysis
419:Lebesgue integration
289:Rules and identities
60:
36740:List of derivatives
36576:History of calculus
36491:Cauchy condensation
36388:Exterior derivative
36345:Lagrange multiplier
36081:Maximum and minimum
35912:Limit of a sequence
35900:Limit of a function
35847:Graph of a function
35827:Continuous function
35717:, Paul L. Fackler,
35341:for the derivation.
35187:The Matrix Cookbook
35108:Felippa, Carlos A.
34505:regression analysis
33901:
33506:-th eigenvector of
32809:
32664:
32357:conjugate transpose
31562:
31187:
29863:
29515:
27761:
20655:
16866:
12428:Denominator layout,
12407:
10479:
6746:
5887:as a column vector.
5503:Hessian formulation
2676:. Notice here that
2309:is identified with
2216:partial derivatives
1991:partial derivatives
1986:functional analysis
1679:
1004:partial derivatives
617:Cauchy condensation
424:Contour integration
150:Fundamental theorem
77:
36673:rational functions
36640:Method of Fluxions
36486:Alternating series
36383:Differential forms
36365:Partial derivative
36325:Divergence theorem
36207:Quadratic integral
35975:Leibniz's notation
35965:Mean value theorem
35950:Partial derivative
35895:Indeterminate form
35730:2012-09-16 at the
35662:MatrixCalculus.org
35547:Statistical Papers
35293:
35245:
35046:
34863:
34643:
34591:
34533:Mathematics portal
34466:
34418:
34375:
34325:
34280:
34232:
34189:
34141:
34086:
34031:
33982:
33938:
33899:
33878:
33808:
33785:
33751:
33719:
33676:
33644:
33613:
33562:
33492:
33388:
33351:
33346:
33147:
33117:
33068:
33031:
32957:
32934:
32897:
32795:
32777:
32728:
32650:
32632:
32584:
32489:
32437:
32396:
32341:
32272:
32221:
32182:
32134:
32069:
32017:
31952:
31900:
31841:
31797:
31760:
31713:
31683:
31632:
31611:
31560:
31541:
31484:
31431:
31319:
31272:
31234:
31185:
31152:
31098:
31016:
30953:
30788:
30684:
30595:
30525:
30283:
30198:
30127:
30050:
29964:
29861:
29837:
29756:
29666:
29590:
29513:
29478:
29417:
29349:
29270:
29116:
28899:
28811:
28738:
28667:
28586:
28496:
28415:
28325:
28250:
28163:
28098:
28008:
27958:
27874:
27831:
27751:
27721:
27697:
27695:
27511:
27509:
27311:
27209:
27112:
26994:
26892:
26809:
26670:
26575:
26539:
26510:
26420:
26372:
26339:
26256:
26185:
26129:
26060:
25991:
25937:
25854:
25806:
25773:
25694:
25613:
25540:
25480:
25432:
25397:
25323:
25278:
25248:
25177:
25074:
24988:
24893:
24839:
24800:
24725:
24655:
24571:
24466:
24426:
24400:
24271:
24178:
24122:
24036:
23983:
23923:
23835:
23812:
23782:
23646:
23616:
23593:
23475:
23419:
23380:
23222:
23165:
23074:
22981:
22883:
22860:
22799:
22717:
22618:
22507:
22423:
22322:
22195:
22160:
22125:
22035:
22000:
21965:
21872:
21826:Both forms assume
21812:
21698:
21599:
21518:
21420:
21341:
21272:
21202:
21131:
21052:
20987:
20899:
20856:
20780:
20757:
20727:
20645:
20615:
20592:
20468:(numerator layout)
20458:
20307:
20305:
19063:
18985:
18983:
18765:
18713:
18651:
18548:
18467:
18372:
18262:
18152:
18064:
17989:
17914:
17842:
17739:
17636:
17539:
17474:
17384:
17329:
17261:
17209:
17164:
17087:
17044:
16967:
16944:
16856:
16826:
16802:
16744:
16673:
16597:
16467:
16337:
16180:
16100:
16013:
15904:
15863:
15814:
15773:
15719:
15590:
15564:
15531:
15387:
15361:
15252:
15214:
15115:
15084:
15046:
14959:
14906:
14846:
14769:
14734:
14699:
14610:
14587:
14557:
14461:
14342:
14319:
14289:
14218:
14150:
14052:
13984:
13874:
13718:
13650:
13569:
13501:
13408:
13277:
13179:
13100:
13031:
12961:
12890:
12811:
12746:
12658:
12615:
12538:
12515:
12485:
12397:
12349:
12322:
12210:
12098:
12013:
11936:
11859:
11785:
11716:
11624:
11570:
11523:
11442:
11360:
11285:
11204:
11152:
11107:
11028:
10983:
10903:
10880:
10850:
10778:
10748:
10725:
10660:
10637:
10588:
10565:
10469:
10437:
10194:
10192:
10181:
9814:
9397:
9255:
9088:
9086:
9075:
8848:
8467:
8465:
8454:
8087:
7670:
7522:
7335:
7291:
7237:
7144:
7090:
7039:
6946:
6897:
6854:
6744:
6725:maximum likelihood
6700:
6661:
6541:
6485:
6440:
6389:
6344:
6292:
6226:
6184:
6143:
6094:
6052:
6005:
5966:
5924:
5877:
5838:
5796:
5748:
5706:
5674:When handling the
5654:
5612:(the denominator).
5596:
5551:
5483:Denominator layout
5463:
5422:
5400:
5254:
5215:Layout conventions
5177:
5059:
4959:
4950:
4546:
4537:
4147:layout conventions
4109:
3983:
3974:
3550:
3536:
3461:
3447:
3352:
3284:
3186:electric potential
3163:
3115:
3001:of a scalar field
2980:
2971:
2967:
2931:
2900:
2816:
2802:
2666:
2596:
2587:
2423:
2409:
2327:layout conventions
2293:-vectors with the
2250:Einstein summation
2224:layout conventions
1982:Fréchet derivative
1970:layout conventions
1940:
1893:
1851:
1806:
1766:
1726:
1677:
1635:
1618:
1614:
1583:
1552:
1445:
1419:
1387:
1360:
1321:
1145:
1067:layout conventions
789:Partial derivative
718:generalized Stokes
612:Alternating series
493:Reduction formulae
468:tangent half-angle
455:Cylindrical shells
378:Integral transform
373:Lists of integrals
177:Mean value theorem
132:
63:
32:geometric calculus
36833:
36832:
36759:Complex calculus
36748:
36747:
36629:Law of Continuity
36561:Natural logarithm
36546:Bernoulli numbers
36537:Special functions
36496:Direct comparison
36360:Multiple integral
36234:Integral equation
36130:Integral calculus
36061:Stationary points
36035:Other techniques
35980:Newton's notation
35945:Second derivative
35837:Finite difference
35764:Matrix Identities
35670:, an open-source
35619:978-0-471-75156-4
35592:978-0-511-64796-3
35473:978-1-4020-3418-3
35395:by S Adler (IAS)
35371:978-3-540-68935-5
35288:
35240:
35041:
35012:
34983:
34954:
34907:
34858:
34829:
34809:
34796:
34773:
34755:
34726:
34708:
34613:is the length of
34556:Tensor derivative
34492:, especially the
34479:
34478:
34456:
34365:
34270:
34179:
34069:
33974:
33776:
33710:
33518:is the matrix of
33368:In the last row,
33364:
33363:
33317:
33135:
32925:
31916:Kronecker product
31635:{\displaystyle 0}
31554:
31553:
31165:
31164:
31093:
30948:
30781:
30749:
30677:
30655:
30590:
30503:
30442:
30379:
30278:
30191:
30122:
30043:
29959:
29850:
29849:
29827:
29802:
29751:
29664:
29639:
29585:
29491:
29490:
29412:
29265:
29094:
29057:
29032:
28994:
28957:
28894:
28794:
28733:
28657:
28632:
28581:
28486:
28461:
28410:
28318:
28293:
28245:
28161:
28136:
28093:
28001:
27953:
27872:
27826:
27749:
27710:
27709:
27306:
26989:
26665:
26505:
26334:
26124:
25932:
25768:
25535:
25392:
25243:
24983:
24795:
24566:
24395:
24350:
24117:
23918:
23777:
23726:
23588:
23546:
23375:
23220:
23160:
23072:
23033:
22976:
22855:
22613:
22317:
22120:
21960:
21867:
21805:
21763:
21691:
21659:
21594:
21516:
21494:
21465:
21415:
21339:
21317:
21267:
21200:
21172:
21126:
21050:
21025:
20982:
20897:
20851:
20722:
20643:
20532:
20384:
19058:
18763:
18711:
18664:
18663:
18634:
18605:
18546:
18505:
18462:
18370:
18335:
18300:
18260:
18238:
18203:
18147:
18062:
18027:
17987:
17965:
17909:
17830:
17777:
17737:
17679:
17631:
17537:
17512:
17469:
17372:
17324:
17247:
17207:
17159:
17085:
17039:
16939:
16854:
16815:
16814:
16800:
16742:
16668:
16591:
16581:
16574:
16564:
16551:
16537:
16527:
16520:
16504:
16490:
16464:
16457:
16440:
16430:
16423:
16410:
16397:
16379:
16369:
16362:
16332:
16328:
16312:
16302:
16295:
16285:
16268:
16258:
16251:
16177:
16159:
16151:
16135:
16127:
16086:
16078:
16062:
16054:
16036:
16008:
16004:
15991:
15978:
15970:
15957:
15902:
15856:
15812:
15771:
15714:
15675:
15526:
15484:
15445:
15356:
15209:
15041:
14841:
14694:
14552:
14513:
14456:
14415:
14284:
14216:
14189:
14131:
14094:
14050:
14023:
13982:
13931:
13869:
13825:
13716:
13689:
13643:
13611:
13567:
13540:
13499:
13460:
13403:
13364:
13275:
13253:
13224:
13174:
13098:
13076:
13026:
12959:
12931:
12885:
12809:
12784:
12741:
12656:
12610:
12480:
12417:Numerator layout,
12377:
12335:
12334:
12320:
12285:
12250:
12208:
12184:
12149:
12093:
12011:
11976:
11934:
11910:
11854:
11783:
11756:
11711:
11610:
11568:
11518:
11428:
11403:
11358:
11328:
11280:
11190:
11150:
11102:
11026:
10978:
10845:
10720:
10632:
10560:
10467:
10177:
10140:
10108:
10052:
10015:
9986:
9955:
9918:
9889:
9847:
9810:
9769:
9733:
9673:
9632:
9596:
9558:
9517:
9481:
9432:
9393:
9359:
9330:
9288:
9251:
9213:
9182:
9140:
8844:
8807:
8775:
8719:
8682:
8653:
8622:
8585:
8556:
8514:
8450:
8413:
8381:
8325:
8288:
8259:
8228:
8191:
8162:
8120:
8083:
8042:
8006:
7946:
7905:
7869:
7831:
7790:
7754:
7705:
7666:
7628:
7597:
7555:
7518:
7484:
7455:
7413:
7367:
7366:
7333:
7289:
7235:
7142:
7088:
7037:
6944:
6895:
6852:
6698:
6659:
6634:
6607:
6582:
6536:
6504:Use the notation
6483:
6438:
6411:, which lays out
6387:
6342:
6315:, which lays out
6287:
6221:
6182:
6138:
6089:
6050:
6003:
5964:
5919:
5875:
5836:
5791:
5743:
5704:
5649:
5594:
5546:
5461:
5398:
5306:matrix, i.e. the
5252:
5192:estimation theory
5162:
5057:
4946:
4909:
4877:
4821:
4784:
4755:
4724:
4687:
4658:
4620:
4533:
4496:
4464:
4408:
4371:
4342:
4311:
4274:
4245:
4207:
4173:and is given (in
4096:
3970:
3929:
3893:
3833:
3792:
3756:
3718:
3677:
3641:
3596:
3560:, is written (in
3342:
3149:
3111:
3073:
2966:
2930:
2899:
2860:
2826:, is written (in
2664:
2583:
2545:
2514:
2476:
2302:, and the scalar
1957:
1956:
1938:
1891:
1849:
1804:
1764:
1724:
1628:
1613:
1582:
1551:
1499:
1351:
1309:
1297:
1260:
1248:
1211:
1199:
1059:estimation theory
984:
983:
864:
863:
826:
825:
794:Multiple integral
730:
729:
634:
633:
601:Direct comparison
572:Convergence tests
510:
509:
483:Partial fractions
350:
349:
260:Second derivative
18:Matrix derivative
16:(Redirected from
36863:
36763:Contour integral
36661:
36660:
36511:Limit comparison
36420:Types of series
36379:Advanced topics
36370:Surface integral
36214:Trapezoidal rule
36153:Basic properties
36148:Riemann integral
36096:Taylor's theorem
35822:Concave function
35817:Binomial theorem
35794:
35787:
35780:
35771:
35770:
35701:, Mike Brookes,
35644:
35635:(9): 2200–2206.
35623:
35604:
35567:
35566:
35542:
35536:
35535:
35533:
35509:
35503:
35502:
35484:
35478:
35477:
35459:
35453:
35452:
35430:
35424:
35423:
35422:. 9783540176510.
35402:
35396:
35393:Unpublished memo
35390:
35384:
35383:
35348:
35342:
35335:
35329:
35328:
35326:
35324:
35318:
35309:
35303:
35302:
35300:
35299:
35294:
35289:
35287:
35286:
35277:
35269:
35260:
35254:
35252:
35251:
35246:
35241:
35239:
35231:
35230:
35221:
35212:
35206:
35204:
35202:
35193:. Archived from
35192:
35181:
35142:
35128:
35126:
35124:
35114:
35105:
35099:
35098:
35096:
35094:
35082:
35056:
35055:
35053:
35052:
35047:
35042:
35040:
35032:
35018:
35013:
35011:
35003:
34989:
34984:
34982:
34974:
34960:
34955:
34953:
34945:
34913:
34908:
34906:
34898:
34881:
34872:
34870:
34869:
34864:
34859:
34857:
34849:
34835:
34830:
34828:
34820:
34812:
34810:
34802:
34797:
34795:
34787:
34779:
34774:
34772:
34761:
34756:
34754:
34746:
34729:
34727:
34725:
34714:
34709:
34707:
34699:
34682:
34673:
34666:
34660:
34658:
34652:
34650:
34649:
34644:
34642:
34629:
34620:
34618:
34612:
34608:
34600:
34598:
34597:
34592:
34590:
34577:
34546:Product integral
34535:
34530:
34529:
34475:
34473:
34472:
34467:
34465:
34457:
34455:
34447:
34446:
34437:
34427:
34425:
34424:
34419:
34410:
34402:
34384:
34382:
34381:
34376:
34374:
34366:
34364:
34363:
34354:
34353:
34344:
34334:
34332:
34331:
34326:
34324:
34315:
34307:
34289:
34287:
34286:
34281:
34279:
34271:
34269:
34261:
34260:
34251:
34241:
34239:
34238:
34233:
34224:
34216:
34198:
34196:
34195:
34190:
34188:
34180:
34178:
34177:
34168:
34160:
34150:
34148:
34147:
34142:
34137:
34128:
34095:
34093:
34092:
34087:
34085:
34084:
34079:
34070:
34068:
34067:
34058:
34050:
34040:
34038:
34037:
34032:
34030:
34022:
34021:
34016:
33991:
33989:
33988:
33983:
33975:
33973:
33965:
33957:
33947:
33945:
33944:
33939:
33902:
33898:
33889:
33887:
33885:
33884:
33879:
33877:
33876:
33871:
33865:
33864:
33849:
33848:
33843:
33837:
33836:
33831:
33817:
33815:
33814:
33809:
33807:
33806:
33801:
33795:
33794:
33784:
33772:
33760:
33758:
33757:
33752:
33750:
33749:
33744:
33735:
33734:
33718:
33703:
33685:
33683:
33682:
33677:
33653:
33651:
33650:
33645:
33640:
33622:
33620:
33619:
33614:
33612:
33611:
33599:
33598:
33586:
33571:
33569:
33568:
33563:
33561:
33560:
33552:
33546:
33541:
33533:
33517:
33511:
33505:
33501:
33499:
33498:
33493:
33491:
33490:
33478:
33477:
33469:
33460:
33459:
33447:
33436:
33435:
33423:
33422:
33417:
33397:
33395:
33394:
33389:
33387:
33386:
33360:
33358:
33357:
33352:
33350:
33349:
33343:
33342:
33330:
33329:
33318:
33316:
33315:
33314:
33302:
33301:
33291:
33287:
33286:
33265:
33264:
33248:
33242:
33241:
33229:
33228:
33214:
33213:
33201:
33185:
33184:
33179:
33170:
33159:
33158:
33153:
33146:
33126:
33124:
33123:
33118:
33113:
33109:
33105:
33077:
33075:
33074:
33069:
33067:
33066:
33046:
33040:
33038:
33037:
33032:
33030:
33029:
33024:
33018:
33017:
33002:
33001:
32996:
32990:
32989:
32984:
32966:
32964:
32963:
32958:
32956:
32955:
32950:
32944:
32943:
32933:
32921:
32906:
32904:
32903:
32898:
32889:
32888:
32876:
32864:
32853:
32842:
32841:
32829:
32817:
32808:
32803:
32786:
32784:
32783:
32778:
32773:
32769:
32768:
32737:
32735:
32734:
32729:
32720:
32719:
32718:
32690:
32679:
32678:
32677:
32663:
32658:
32641:
32639:
32638:
32633:
32628:
32624:
32623:
32622:
32593:
32591:
32590:
32585:
32583:
32582:
32565:
32556:
32545:
32544:
32539:
32532:
32521:
32498:
32496:
32495:
32490:
32485:
32481:
32480:
32475:
32453:
32446:
32444:
32443:
32438:
32436:
32435:
32434:
32424:
32405:
32403:
32402:
32397:
32392:
32388:
32387:
32386:
32380:
32350:
32348:
32347:
32342:
32340:
32339:
32331:
32325:
32321:
32320:
32307:
32306:
32298:
32281:
32279:
32278:
32273:
32268:
32264:
32263:
32255:
32230:
32228:
32227:
32222:
32220:
32219:
32210:
32191:
32189:
32188:
32183:
32178:
32174:
32173:
32168:
32143:
32141:
32140:
32135:
32130:
32116:
32108:
32097:
32078:
32076:
32075:
32070:
32062:
32054:
32033:Hadamard product
32026:
32024:
32023:
32018:
32013:
31999:
31991:
31980:
31961:
31959:
31958:
31953:
31945:
31937:
31909:
31907:
31906:
31901:
31896:
31885:
31877:
31869:
31850:
31848:
31847:
31842:
31834:
31829:
31806:
31804:
31803:
31798:
31796:
31785:
31769:
31767:
31766:
31761:
31753:
31745:
31722:
31720:
31719:
31714:
31712:
31692:
31690:
31689:
31684:
31676:
31654:
31641:
31639:
31638:
31633:
31620:
31618:
31617:
31612:
31604:
31585:
31563:
31559:
31550:
31548:
31547:
31542:
31540:
31536:
31535:
31527:
31526:
31518:
31493:
31491:
31490:
31485:
31477:
31472:
31467:
31440:
31438:
31437:
31432:
31427:
31416:
31390:
31386:
31385:
31377:
31376:
31368:
31351:
31346:
31341:
31328:
31326:
31325:
31320:
31312:
31307:
31302:
31281:
31279:
31278:
31273:
31268:
31243:
31241:
31240:
31235:
31224:
31188:
31184:
31179:
31161:
31159:
31158:
31153:
31151:
31147:
31146:
31145:
31144:
31131:
31107:
31105:
31104:
31099:
31094:
31092:
31084:
31083:
31079:
31078:
31077:
31050:
31040:
31034:
31025:
31023:
31022:
31017:
31015:
31011:
31007:
30996:
30992:
30986:
30962:
30960:
30959:
30954:
30949:
30947:
30939:
30932:
30921:
30903:
30892:
30878:
30864:
30853:
30845:
30837:
30829:
30823:
30812:
30806:
30797:
30795:
30794:
30789:
30787:
30783:
30782:
30780:
30772:
30771:
30762:
30760:
30759:
30754:
30750:
30748:
30747:
30738:
30734:
30719:
30693:
30691:
30690:
30685:
30683:
30679:
30678:
30676:
30668:
30667:
30658:
30656:
30654:
30653:
30644:
30640:
30625:
30604:
30602:
30601:
30596:
30591:
30589:
30581:
30577:
30562:
30552:
30534:
30532:
30531:
30526:
30524:
30520:
30519:
30515:
30514:
30509:
30505:
30504:
30502:
30494:
30493:
30484:
30482:
30481:
30473:
30448:
30444:
30443:
30441:
30433:
30432:
30423:
30421:
30420:
30412:
30398:
30397:
30385:
30381:
30380:
30378:
30377:
30376:
30363:
30362:
30357:
30356:
30346:
30344:
30343:
30335:
30313:
30309:
30292:
30290:
30289:
30284:
30279:
30277:
30276:
30275:
30262:
30261:
30256:
30251:
30246:
30245:
30235:
30225:
30207:
30205:
30204:
30199:
30197:
30193:
30192:
30190:
30182:
30181:
30172:
30170:
30169:
30161:
30136:
30134:
30133:
30128:
30123:
30121:
30113:
30112:
30107:
30102:
30087:
30077:
30059:
30057:
30056:
30051:
30049:
30045:
30044:
30042:
30034:
30033:
30024:
30022:
30021:
30013:
29996:
29991:
29986:
29973:
29971:
29970:
29965:
29960:
29958:
29950:
29949:
29944:
29939:
29930:
29920:
29902:
29896:
29886:
29880:
29864:
29860:
29846:
29844:
29843:
29838:
29836:
29828:
29826:
29818:
29817:
29808:
29803:
29801:
29793:
29792:
29783:
29778:
29765:
29763:
29762:
29757:
29752:
29750:
29742:
29738:
29730:
29718:
29708:
29693:
29675:
29673:
29672:
29667:
29665:
29663:
29655:
29654:
29645:
29640:
29638:
29637:
29628:
29624:
29609:
29599:
29597:
29596:
29591:
29586:
29584:
29576:
29572:
29557:
29547:
29516:
29512:
29487:
29485:
29484:
29479:
29477:
29472:
29471:
29470:
29454:
29453:
29452:
29439:
29426:
29424:
29423:
29418:
29413:
29411:
29403:
29402:
29401:
29400:
29383:
29373:
29367:
29358:
29356:
29355:
29350:
29348:
29340:
29329:
29325:
29313:
29302:
29298:
29292:
29279:
29277:
29276:
29271:
29266:
29264:
29256:
29252:
29241:
29231:
29220:
29206:
29192:
29181:
29173:
29165:
29157:
29151:
29140:
29134:
29125:
29123:
29122:
29117:
29115:
29114:
29106:
29100:
29096:
29095:
29093:
29085:
29084:
29075:
29073:
29072:
29064:
29058:
29056:
29048:
29047:
29038:
29033:
29031:
29017:
29016:
29011:
29010:
29000:
28995:
28993:
28985:
28984:
28975:
28973:
28972:
28964:
28958:
28956:
28948:
28947:
28938:
28931:
28930:
28922:
28908:
28906:
28905:
28900:
28895:
28893:
28879:
28878:
28877:
28869:
28863:
28862:
28852:
28842:
28820:
28818:
28817:
28812:
28810:
28809:
28801:
28795:
28793:
28785:
28784:
28775:
28773:
28772:
28764:
28747:
28745:
28744:
28739:
28734:
28732:
28724:
28723:
28722:
28714:
28704:
28694:
28676:
28674:
28673:
28668:
28666:
28658:
28656:
28648:
28647:
28638:
28633:
28631:
28623:
28622:
28613:
28608:
28595:
28593:
28592:
28587:
28582:
28580:
28572:
28568:
28560:
28548:
28538:
28523:
28505:
28503:
28502:
28497:
28495:
28487:
28485:
28477:
28476:
28467:
28462:
28460:
28452:
28451:
28442:
28437:
28424:
28422:
28421:
28416:
28411:
28409:
28401:
28397:
28389:
28377:
28367:
28352:
28334:
28332:
28331:
28326:
28324:
28319:
28317:
28309:
28308:
28299:
28294:
28292:
28284:
28283:
28274:
28272:
28259:
28257:
28256:
28251:
28246:
28244:
28236:
28232:
28227:
28215:
28205:
28190:
28172:
28170:
28169:
28164:
28162:
28160:
28152:
28151:
28142:
28137:
28135:
28127:
28126:
28117:
28107:
28105:
28104:
28099:
28094:
28092:
28084:
28080:
28072:
28060:
28050:
28035:
28017:
28015:
28014:
28009:
28007:
28002:
28000:
27992:
27991:
27982:
27980:
27967:
27965:
27964:
27959:
27954:
27952:
27944:
27943:
27928:
27918:
27898:
27892:
27883:
27881:
27880:
27875:
27873:
27871:
27863:
27862:
27853:
27840:
27838:
27837:
27832:
27827:
27825:
27817:
27816:
27804:
27794:
27776:
27762:
27760:
27758:
27757:
27752:
27750:
27748:
27740:
27739:
27730:
27720:
27706:
27704:
27703:
27698:
27696:
27692:
27691:
27685:
27684:
27676:
27672:
27668:
27667:
27658:
27657:
27642:
27638:
27637:
27626:
27622:
27617:
27616:
27608:
27604:
27597:
27596:
27581:
27571:
27570:
27564:
27560:
27559:
27554:
27549:
27548:
27547:
27520:
27518:
27517:
27512:
27510:
27506:
27505:
27499:
27498:
27497:
27488:
27487:
27477:
27476:
27468:
27464:
27460:
27459:
27450:
27449:
27433:
27429:
27424:
27420:
27419:
27409:
27408:
27400:
27396:
27389:
27388:
27371:
27370:
27364:
27360:
27359:
27354:
27349:
27348:
27347:
27320:
27318:
27317:
27312:
27307:
27305:
27304:
27295:
27294:
27289:
27284:
27279:
27278:
27277:
27267:
27258:
27248:is non-symmetric
27247:
27240:
27233:
27227:
27218:
27216:
27215:
27210:
27208:
27207:
27199:
27195:
27188:
27187:
27172:
27164:
27160:
27159:
27154:
27149:
27148:
27147:
27121:
27119:
27118:
27113:
27111:
27110:
27109:
27100:
27099:
27089:
27088:
27080:
27076:
27072:
27071:
27062:
27061:
27046:
27042:
27041:
27036:
27031:
27030:
27029:
27003:
27001:
27000:
26995:
26990:
26988:
26987:
26978:
26977:
26973:
26972:
26967:
26962:
26961:
26960:
26941:
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8911:
8910:
8896:
8895:
8868:
8853:
8852:
8845:
8843:
8835:
8834:
8833:
8817:
8808:
8806:
8798:
8797:
8796:
8780:
8776:
8774:
8766:
8765:
8764:
8748:
8720:
8718:
8710:
8709:
8708:
8692:
8683:
8681:
8673:
8672:
8671:
8658:
8654:
8652:
8644:
8643:
8642:
8629:
8623:
8621:
8613:
8612:
8611:
8595:
8586:
8584:
8576:
8575:
8574:
8561:
8557:
8555:
8547:
8546:
8545:
8532:
8515:
8513:
8505:
8504:
8495:
8476:
8474:
8473:
8468:
8466:
8459:
8458:
8451:
8449:
8448:
8447:
8431:
8423:
8414:
8412:
8411:
8410:
8394:
8386:
8382:
8380:
8379:
8378:
8362:
8354:
8326:
8324:
8323:
8322:
8306:
8298:
8289:
8287:
8286:
8285:
8272:
8264:
8260:
8258:
8257:
8256:
8243:
8235:
8229:
8227:
8226:
8225:
8209:
8201:
8192:
8190:
8189:
8188:
8175:
8167:
8163:
8161:
8160:
8159:
8146:
8138:
8121:
8119:
8118:
8109:
8101:
8092:
8091:
8084:
8082:
8081:
8080:
8067:
8066:
8065:
8052:
8043:
8041:
8040:
8039:
8026:
8025:
8024:
8011:
8007:
8005:
8004:
8003:
7990:
7989:
7988:
7975:
7947:
7945:
7944:
7943:
7930:
7929:
7928:
7915:
7906:
7904:
7903:
7902:
7889:
7888:
7887:
7874:
7870:
7868:
7867:
7866:
7853:
7852:
7851:
7838:
7832:
7830:
7829:
7828:
7815:
7814:
7813:
7800:
7791:
7789:
7788:
7787:
7774:
7773:
7772:
7759:
7755:
7753:
7752:
7751:
7738:
7737:
7736:
7723:
7706:
7704:
7703:
7694:
7693:
7684:
7675:
7674:
7667:
7665:
7657:
7656:
7655:
7642:
7629:
7627:
7619:
7618:
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7604:
7598:
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7588:
7587:
7586:
7573:
7556:
7554:
7546:
7545:
7536:
7527:
7526:
7519:
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7516:
7515:
7502:
7494:
7485:
7483:
7482:
7481:
7468:
7460:
7456:
7454:
7453:
7452:
7439:
7431:
7414:
7412:
7411:
7402:
7394:
7362:
7344:
7342:
7341:
7336:
7334:
7332:
7331:
7322:
7321:
7312:
7300:
7298:
7297:
7292:
7290:
7288:
7287:
7278:
7277:
7268:
7257:
7246:
7244:
7243:
7238:
7236:
7234:
7233:
7224:
7216:
7202:
7191:
7180:
7166:
7153:
7151:
7150:
7145:
7143:
7141:
7140:
7131:
7130:
7121:
7110:
7099:
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7091:
7089:
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7086:
7077:
7076:
7067:
7054:
7048:
7046:
7045:
7040:
7038:
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7035:
7026:
7018:
7004:
6993:
6966:
6955:
6953:
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6945:
6943:
6935:
6934:
6925:
6912:
6906:
6904:
6903:
6898:
6896:
6894:
6886:
6885:
6876:
6863:
6861:
6860:
6855:
6853:
6851:
6843:
6835:
6822:
6791:
6781:
6772:
6765:
6757:
6747:
6743:
6712:parametric curve
6709:
6707:
6706:
6701:
6699:
6697:
6689:
6688:
6679:
6670:
6668:
6667:
6662:
6660:
6658:
6657:
6648:
6640:
6635:
6633:
6632:
6623:
6622:
6613:
6608:
6606:
6598:
6597:
6588:
6583:
6581:
6580:
6571:
6563:
6550:
6548:
6547:
6542:
6537:
6535:
6534:
6530:
6520:
6512:
6500:
6494:
6492:
6491:
6486:
6484:
6482:
6481:
6472:
6464:
6455:
6449:
6447:
6446:
6441:
6439:
6437:
6429:
6428:
6419:
6404:
6398:
6396:
6395:
6390:
6388:
6386:
6385:
6376:
6368:
6359:
6353:
6351:
6350:
6345:
6343:
6341:
6333:
6332:
6323:
6307:
6301:
6299:
6298:
6293:
6288:
6286:
6278:
6277:
6268:
6259:
6253:
6247:
6241:
6235:
6233:
6232:
6227:
6222:
6220:
6212:
6211:
6202:
6193:
6191:
6190:
6185:
6183:
6181:
6180:
6171:
6163:
6152:
6150:
6149:
6144:
6139:
6137:
6136:
6127:
6126:
6117:
6103:
6101:
6100:
6095:
6090:
6088:
6080:
6079:
6070:
6061:
6059:
6058:
6053:
6051:
6049:
6048:
6044:
6034:
6026:
6015:as a row vector.
6014:
6012:
6011:
6006:
6004:
6002:
5994:
5993:
5984:
5975:
5973:
5972:
5967:
5965:
5963:
5962:
5953:
5945:
5933:
5931:
5930:
5925:
5920:
5918:
5917:
5908:
5907:
5898:
5886:
5884:
5883:
5878:
5876:
5874:
5866:
5865:
5856:
5847:
5845:
5844:
5839:
5837:
5835:
5834:
5825:
5817:
5805:
5803:
5802:
5797:
5792:
5790:
5789:
5780:
5779:
5770:
5757:
5755:
5754:
5749:
5744:
5742:
5734:
5733:
5724:
5715:
5713:
5712:
5707:
5705:
5703:
5702:
5693:
5685:
5669:
5663:
5661:
5660:
5655:
5650:
5648:
5647:
5643:
5633:
5632:
5623:
5611:
5605:
5603:
5602:
5597:
5595:
5593:
5592:
5583:
5582:
5573:
5560:
5558:
5557:
5552:
5547:
5545:
5544:
5535:
5527:
5496:
5490:
5478:
5472:
5470:
5469:
5464:
5462:
5460:
5459:
5450:
5449:
5440:
5431:
5429:
5428:
5423:
5421:
5409:
5407:
5406:
5401:
5399:
5397:
5396:
5387:
5386:
5377:
5368:
5354:
5348:
5342:
5335:Numerator layout
5329:
5323:
5317:
5311:
5305:
5295:
5281:
5275:
5269:
5263:
5261:
5260:
5255:
5253:
5251:
5250:
5241:
5240:
5231:
5186:
5184:
5183:
5178:
5173:
5169:
5168:
5163:
5161:
5160:
5151:
5143:
5124:
5123:
5122:
5102:
5096:
5090:
5068:
5066:
5065:
5060:
5058:
5056:
5055:
5046:
5042:
5027:
5019:
5008:
5007:
5006:
4968:
4966:
4965:
4960:
4955:
4954:
4947:
4945:
4944:
4943:
4927:
4919:
4910:
4908:
4907:
4906:
4890:
4882:
4878:
4876:
4875:
4874:
4858:
4850:
4822:
4820:
4819:
4818:
4802:
4794:
4785:
4783:
4782:
4781:
4768:
4760:
4756:
4754:
4753:
4752:
4739:
4731:
4725:
4723:
4722:
4721:
4705:
4697:
4688:
4686:
4685:
4684:
4671:
4663:
4659:
4657:
4656:
4655:
4642:
4634:
4621:
4619:
4618:
4609:
4601:
4585:
4579:
4569:
4560:Scalar-by-matrix
4555:
4553:
4552:
4547:
4542:
4541:
4534:
4532:
4524:
4523:
4522:
4506:
4497:
4495:
4487:
4486:
4485:
4469:
4465:
4463:
4455:
4454:
4453:
4437:
4409:
4407:
4399:
4398:
4397:
4381:
4372:
4370:
4362:
4361:
4360:
4347:
4343:
4341:
4333:
4332:
4331:
4318:
4312:
4310:
4302:
4301:
4300:
4284:
4275:
4273:
4265:
4264:
4263:
4250:
4246:
4244:
4236:
4235:
4234:
4221:
4208:
4206:
4198:
4197:
4188:
4169:is known as the
4168:
4162:
4153:Matrix-by-scalar
4118:
4116:
4115:
4110:
4105:
4097:
4095:
4094:
4085:
4084:
4075:
4067:
4059:
4044:
4038:
4032:
4011:
4005:
3992:
3990:
3989:
3984:
3979:
3978:
3971:
3969:
3968:
3967:
3954:
3953:
3952:
3939:
3930:
3928:
3927:
3926:
3913:
3912:
3911:
3898:
3894:
3892:
3891:
3890:
3877:
3876:
3875:
3862:
3834:
3832:
3831:
3830:
3817:
3816:
3815:
3802:
3793:
3791:
3790:
3789:
3776:
3775:
3774:
3761:
3757:
3755:
3754:
3753:
3740:
3739:
3738:
3725:
3719:
3717:
3716:
3715:
3702:
3701:
3700:
3687:
3678:
3676:
3675:
3674:
3661:
3660:
3659:
3646:
3642:
3640:
3639:
3638:
3625:
3624:
3623:
3610:
3597:
3595:
3594:
3585:
3584:
3575:
3559:
3557:
3556:
3551:
3549:
3548:
3547:
3541:
3540:
3533:
3532:
3516:
3515:
3504:
3503:
3482:
3470:
3468:
3467:
3462:
3460:
3459:
3458:
3452:
3451:
3444:
3443:
3427:
3426:
3415:
3414:
3393:
3370:Vector-by-vector
3361:
3359:
3358:
3353:
3348:
3343:
3341:
3340:
3331:
3323:
3315:
3314:
3313:
3293:
3291:
3290:
3285:
3283:
3272:
3252:
3244:
3239:
3238:
3237:
3217:
3211:
3205:
3172:
3170:
3169:
3164:
3162:
3161:
3160:
3154:
3150:
3148:
3147:
3138:
3130:
3120:
3119:
3112:
3110:
3109:
3108:
3095:
3087:
3074:
3072:
3071:
3070:
3057:
3049:
3020:
3014:
2989:
2987:
2986:
2981:
2976:
2975:
2968:
2965:
2964:
2963:
2950:
2942:
2932:
2929:
2928:
2927:
2914:
2906:
2901:
2898:
2897:
2896:
2883:
2875:
2861:
2859:
2858:
2849:
2841:
2825:
2823:
2822:
2817:
2815:
2814:
2813:
2807:
2806:
2799:
2798:
2782:
2781:
2770:
2769:
2748:
2736:
2720:Scalar-by-vector
2689:
2675:
2673:
2672:
2667:
2665:
2663:
2655:
2654:
2645:
2636:
2625:is known as the
2624:
2618:
2605:
2603:
2602:
2597:
2592:
2591:
2584:
2582:
2574:
2573:
2572:
2559:
2546:
2544:
2536:
2535:
2534:
2521:
2515:
2513:
2505:
2504:
2503:
2490:
2477:
2475:
2467:
2466:
2457:
2441:
2432:
2430:
2429:
2424:
2422:
2421:
2420:
2414:
2413:
2406:
2405:
2389:
2388:
2377:
2376:
2355:
2333:Vector-by-scalar
2314:
2308:
2301:
2292:
2288:
2207:
2194:
2192:
2184:
2172:
2160:
2154:
2148:
2142:
2136:
2129:
2123:
2117:
2107:
2096:
2090:
2084:
2078:
2072:
2066:
2053:
2025:Gradient descent
1949:
1947:
1946:
1941:
1939:
1937:
1936:
1927:
1919:
1902:
1900:
1899:
1894:
1892:
1890:
1889:
1880:
1879:
1870:
1860:
1858:
1857:
1852:
1850:
1848:
1847:
1838:
1830:
1815:
1813:
1812:
1807:
1805:
1803:
1795:
1794:
1785:
1775:
1773:
1772:
1767:
1765:
1763:
1755:
1754:
1745:
1735:
1733:
1732:
1727:
1725:
1723:
1715:
1707:
1680:
1676:
1669:
1659:
1655:
1644:
1642:
1641:
1636:
1631:
1630:
1629:
1623:
1622:
1615:
1612:
1611:
1610:
1597:
1589:
1584:
1581:
1580:
1579:
1566:
1558:
1553:
1550:
1549:
1548:
1535:
1527:
1512:
1511:
1510:
1504:
1500:
1498:
1497:
1488:
1480:
1454:
1452:
1451:
1446:
1444:
1428:
1426:
1425:
1420:
1396:
1394:
1393:
1388:
1386:
1385:
1369:
1367:
1366:
1361:
1359:
1358:
1353:
1352:
1344:
1330:
1328:
1327:
1322:
1317:
1316:
1311:
1310:
1302:
1298:
1296:
1295:
1294:
1281:
1273:
1268:
1267:
1262:
1261:
1253:
1249:
1247:
1246:
1245:
1232:
1224:
1219:
1218:
1213:
1212:
1204:
1200:
1198:
1197:
1196:
1183:
1175:
1154:
1152:
1151:
1146:
1141:
1140:
1128:
1127:
1115:
1114:
1063:machine learning
1038:is preferred in
976:
969:
962:
910:
875:
841:
840:
837:
804:Surface integral
747:
746:
743:
651:
650:
647:
607:Limit comparison
527:
526:
523:
414:Riemann integral
367:
366:
363:
323:L'Hôpital's rule
280:Taylor's theorem
201:
200:
197:
141:
139:
138:
133:
85:
76:
71:
41:
40:
21:
36871:
36870:
36866:
36865:
36864:
36862:
36861:
36860:
36836:
36835:
36834:
36829:
36825:Steinmetz solid
36810:Integration Bee
36744:
36726:
36652:
36594:Colin Maclaurin
36570:
36538:
36532:
36404:
36398:Tensor calculus
36375:Volume integral
36311:
36286:Basic theorems
36249:Vector calculus
36243:
36124:
36091:Newton's method
35926:
35905:One-sided limit
35881:
35862:Rolle's theorem
35852:Linear function
35803:
35798:
35732:Wayback Machine
35695:
35658:
35653:
35648:
35620:
35593:
35576:
35574:Further reading
35571:
35570:
35543:
35539:
35510:
35506:
35499:
35485:
35481:
35474:
35460:
35456:
35449:
35431:
35427:
35420:
35403:
35399:
35391:
35387:
35372:
35349:
35345:
35336:
35332:
35322:
35320:
35316:
35312:Duchi, John C.
35310:
35306:
35282:
35278:
35270:
35268:
35266:
35263:
35262:
35256:
35232:
35226:
35222:
35220:
35218:
35215:
35214:
35208:
35200:
35198:
35197:on 2 March 2010
35190:
35182:
35145:
35122:
35120:
35112:
35106:
35102:
35092:
35090:
35083:
35070:
35065:
35060:
35059:
35033:
35019:
35017:
35004:
34990:
34988:
34975:
34961:
34959:
34946:
34914:
34912:
34899:
34882:
34880:
34878:
34875:
34874:
34850:
34836:
34834:
34821:
34813:
34811:
34801:
34788:
34780:
34778:
34765:
34760:
34747:
34730:
34728:
34718:
34713:
34700:
34683:
34681:
34679:
34676:
34675:
34669:
34667:
34663:
34654:
34638:
34636:
34633:
34632:
34630:
34623:
34614:
34610:
34606:
34586:
34584:
34581:
34580:
34578:
34569:
34564:
34531:
34524:
34521:
34486:
34461:
34448:
34442:
34438:
34436:
34434:
34431:
34430:
34406:
34398:
34393:
34390:
34389:
34370:
34359:
34355:
34349:
34345:
34343:
34341:
34338:
34337:
34320:
34311:
34303:
34298:
34295:
34294:
34275:
34262:
34256:
34252:
34250:
34248:
34245:
34244:
34220:
34212:
34207:
34204:
34203:
34184:
34173:
34169:
34161:
34159:
34157:
34154:
34153:
34133:
34124:
34104:
34101:
34100:
34080:
34075:
34074:
34063:
34059:
34051:
34049:
34047:
34044:
34043:
34026:
34017:
34012:
34011:
34000:
33997:
33996:
33966:
33958:
33956:
33954:
33951:
33950:
33917:
33914:
33913:
33872:
33867:
33866:
33857:
33853:
33844:
33839:
33838:
33832:
33827:
33826:
33824:
33821:
33820:
33819:
33802:
33797:
33796:
33790:
33786:
33780:
33768:
33766:
33763:
33762:
33745:
33740:
33739:
33730:
33726:
33714:
33699:
33691:
33688:
33687:
33662:
33659:
33658:
33636:
33628:
33625:
33624:
33607:
33603:
33591:
33587:
33582:
33577:
33574:
33573:
33553:
33548:
33547:
33542:
33537:
33529:
33527:
33524:
33523:
33513:
33507:
33503:
33483:
33479:
33470:
33465:
33464:
33452:
33448:
33443:
33428:
33424:
33418:
33413:
33412:
33407:
33404:
33403:
33400:Kronecker delta
33379:
33375:
33373:
33370:
33369:
33345:
33344:
33338:
33334:
33325:
33321:
33319:
33310:
33306:
33297:
33293:
33292:
33282:
33278:
33260:
33256:
33249:
33247:
33244:
33243:
33237:
33233:
33224:
33220:
33218:
33209:
33205:
33194:
33187:
33186:
33180:
33175:
33174:
33166:
33154:
33149:
33148:
33139:
33133:
33130:
33129:
33101:
33094:
33090:
33085:
33082:
33081:
33062:
33058:
33056:
33053:
33052:
33042:
33041:
33025:
33020:
33019:
33010:
33006:
32997:
32992:
32991:
32985:
32980:
32979:
32977:
32974:
32973:
32971:
32951:
32946:
32945:
32939:
32935:
32929:
32917:
32915:
32912:
32911:
32881:
32877:
32872:
32860:
32849:
32834:
32830:
32825:
32813:
32804:
32799:
32793:
32790:
32789:
32764:
32757:
32753:
32748:
32745:
32744:
32714:
32698:
32694:
32686:
32673:
32669:
32665:
32659:
32654:
32648:
32645:
32644:
32618:
32617:
32613:
32609:
32604:
32601:
32600:
32566:
32561:
32560:
32552:
32540:
32535:
32534:
32522:
32511:
32505:
32502:
32501:
32476:
32471:
32470:
32466:
32461:
32458:
32457:
32451:
32430:
32429:
32425:
32420:
32412:
32409:
32408:
32382:
32381:
32376:
32375:
32371:
32366:
32363:
32362:
32332:
32327:
32326:
32316:
32312:
32308:
32299:
32294:
32293:
32288:
32285:
32284:
32256:
32251:
32250:
32246:
32241:
32238:
32237:
32215:
32211:
32206:
32198:
32195:
32194:
32169:
32164:
32163:
32159:
32154:
32151:
32150:
32126:
32112:
32104:
32093:
32085:
32082:
32081:
32058:
32050:
32042:
32039:
32038:
32009:
31995:
31987:
31976:
31968:
31965:
31964:
31941:
31933:
31925:
31922:
31921:
31892:
31881:
31873:
31865:
31857:
31854:
31853:
31830:
31825:
31817:
31814:
31813:
31792:
31781:
31776:
31773:
31772:
31749:
31741:
31733:
31730:
31729:
31708:
31699:
31696:
31695:
31672:
31661:
31658:
31657:
31650:
31627:
31624:
31623:
31600:
31592:
31589:
31588:
31581:
31531:
31519:
31514:
31513:
31512:
31508:
31500:
31497:
31496:
31473:
31468:
31463:
31449:
31446:
31445:
31423:
31412:
31381:
31369:
31364:
31363:
31362:
31358:
31347:
31342:
31337:
31335:
31332:
31331:
31308:
31303:
31298:
31290:
31287:
31286:
31264:
31250:
31247:
31246:
31220:
31203:
31200:
31199:
31175:
31172:
31140:
31136:
31132:
31127:
31126:
31122:
31114:
31111:
31110:
31085:
31073:
31069:
31065:
31061:
31051:
31049:
31047:
31044:
31043:
31036:
31030:
31003:
30988:
30987:
30982:
30981:
30977:
30969:
30966:
30965:
30940:
30928:
30917:
30904:
30902:
30900:
30897:
30896:
30886:
30880:
30872:
30866:
30855:
30847:
30839:
30831:
30825:
30814:
30808:
30802:
30773:
30767:
30763:
30761:
30755:
30743:
30739:
30730:
30720:
30718:
30714:
30713:
30712:
30708:
30700:
30697:
30696:
30669:
30663:
30659:
30657:
30649:
30645:
30636:
30626:
30624:
30623:
30619:
30611:
30608:
30607:
30582:
30573:
30563:
30561:
30559:
30556:
30555:
30539:
30510:
30495:
30489:
30485:
30483:
30474:
30469:
30468:
30467:
30463:
30462:
30458:
30434:
30428:
30424:
30422:
30413:
30408:
30407:
30406:
30402:
30393:
30389:
30372:
30368:
30364:
30358:
30352:
30348:
30347:
30345:
30336:
30331:
30330:
30329:
30325:
30318:
30314:
30305:
30301:
30299:
30296:
30295:
30271:
30267:
30263:
30257:
30252:
30247:
30241:
30237:
30236:
30234:
30232:
30229:
30228:
30212:
30183:
30177:
30173:
30171:
30162:
30157:
30156:
30155:
30151:
30143:
30140:
30139:
30114:
30108:
30103:
30098:
30088:
30086:
30084:
30081:
30080:
30064:
30035:
30029:
30025:
30023:
30014:
30009:
30008:
30007:
30003:
29992:
29987:
29982:
29980:
29977:
29976:
29951:
29945:
29940:
29935:
29931:
29929:
29927:
29924:
29923:
29907:
29898:
29892:
29890:
29882:
29876:
29874:
29857:
29832:
29819:
29813:
29809:
29807:
29794:
29788:
29784:
29782:
29774:
29772:
29769:
29768:
29743:
29734:
29726:
29719:
29717:
29715:
29712:
29711:
29695:
29680:
29656:
29650:
29646:
29644:
29633:
29629:
29620:
29610:
29608:
29606:
29603:
29602:
29577:
29568:
29558:
29556:
29554:
29551:
29550:
29534:
29509:
29504:
29499:
29473:
29466:
29462:
29458:
29448:
29444:
29440:
29435:
29433:
29430:
29429:
29404:
29396:
29392:
29388:
29384:
29382:
29380:
29377:
29376:
29369:
29363:
29344:
29336:
29321:
29320:
29309:
29294:
29293:
29288:
29286:
29283:
29282:
29257:
29248:
29237:
29232:
29230:
29228:
29225:
29224:
29214:
29208:
29200:
29194:
29183:
29175:
29167:
29159:
29153:
29142:
29136:
29130:
29107:
29102:
29101:
29086:
29080:
29076:
29074:
29065:
29060:
29059:
29049:
29043:
29039:
29037:
29018:
29012:
29006:
29002:
29001:
28999:
28986:
28980:
28976:
28974:
28965:
28960:
28959:
28949:
28943:
28939:
28937:
28936:
28932:
28923:
28918:
28917:
28915:
28912:
28911:
28880:
28870:
28865:
28864:
28858:
28854:
28853:
28851:
28849:
28846:
28845:
28825:
28802:
28797:
28796:
28786:
28780:
28776:
28774:
28765:
28760:
28759:
28754:
28751:
28750:
28725:
28715:
28710:
28709:
28705:
28703:
28701:
28698:
28697:
28681:
28662:
28649:
28643:
28639:
28637:
28624:
28618:
28614:
28612:
28604:
28602:
28599:
28598:
28573:
28564:
28556:
28549:
28547:
28545:
28542:
28541:
28525:
28510:
28491:
28478:
28472:
28468:
28466:
28453:
28447:
28443:
28441:
28433:
28431:
28428:
28427:
28402:
28393:
28385:
28378:
28376:
28374:
28371:
28370:
28354:
28339:
28320:
28310:
28304:
28300:
28298:
28285:
28279:
28275:
28273:
28268:
28266:
28263:
28262:
28237:
28228:
28223:
28216:
28214:
28212:
28209:
28208:
28192:
28177:
28153:
28147:
28143:
28141:
28128:
28122:
28118:
28116:
28114:
28111:
28110:
28085:
28076:
28068:
28061:
28059:
28057:
28054:
28053:
28037:
28022:
28003:
27993:
27987:
27983:
27981:
27976:
27974:
27971:
27970:
27945:
27933:
27929:
27927:
27925:
27922:
27921:
27905:
27904:
27894:
27888:
27864:
27858:
27854:
27852:
27847:
27844:
27843:
27818:
27812:
27805:
27803:
27801:
27798:
27797:
27781:
27772:
27741:
27735:
27731:
27729:
27727:
27724:
27723:
27717:
27694:
27693:
27687:
27686:
27677:
27663:
27659:
27653:
27649:
27648:
27644:
27643:
27633:
27629:
27628:
27624:
27623:
27621:
27609:
27592:
27588:
27587:
27583:
27582:
27574:
27572:
27566:
27565:
27555:
27550:
27543:
27539:
27538:
27537:
27533:
27529:
27527:
27524:
27523:
27508:
27507:
27501:
27500:
27493:
27489:
27483:
27479:
27478:
27469:
27455:
27451:
27445:
27441:
27440:
27436:
27435:
27431:
27430:
27428:
27415:
27411:
27410:
27401:
27384:
27380:
27379:
27375:
27374:
27372:
27366:
27365:
27355:
27350:
27343:
27339:
27338:
27337:
27333:
27329:
27327:
27324:
27323:
27300:
27296:
27290:
27285:
27280:
27273:
27269:
27268:
27263:
27259:
27257:
27255:
27252:
27251:
27243:
27242:
27236:
27235:
27229:
27223:
27200:
27183:
27179:
27178:
27174:
27173:
27165:
27155:
27150:
27143:
27139:
27138:
27137:
27133:
27128:
27125:
27124:
27105:
27101:
27095:
27091:
27090:
27081:
27067:
27063:
27057:
27053:
27052:
27048:
27047:
27037:
27032:
27025:
27021:
27020:
27019:
27015:
27010:
27007:
27006:
26983:
26979:
26968:
26963:
26956:
26952:
26951:
26950:
26946:
26942:
26940:
26938:
26935:
26934:
26926:
26925:
26919:
26918:
26912:
26906:
26886:
26873:
26868:
26867:
26863:
26862:
26852:
26847:
26840:
26836:
26835:
26834:
26830:
26825:
26822:
26821:
26800:
26795:
26794:
26789:
26784:
26779:
26774:
26769:
26764:
26753:
26749:
26748:
26744:
26729:
26724:
26723:
26713:
26708:
26701:
26697:
26696:
26695:
26691:
26686:
26683:
26682:
26659:
26655:
26644:
26639:
26632:
26628:
26627:
26626:
26622:
26618:
26616:
26614:
26611:
26610:
26602:
26601:
26595:
26589:
26569:
26564:
26563:
26555:
26552:
26551:
26534:
26526:
26523:
26522:
26498:
26493:
26492:
26488:
26477:
26471:
26466:
26465:
26464:
26460:
26450:
26448:
26446:
26443:
26442:
26414:
26404:
26399:
26398:
26394:
26393:
26388:
26385:
26384:
26366:
26361:
26360:
26355:
26352:
26351:
26328:
26324:
26313:
26307:
26302:
26301:
26300:
26296:
26286:
26284:
26282:
26279:
26278:
26250:
26237:
26232:
26231:
26227:
26226:
26216:
26211:
26210:
26206:
26201:
26198:
26197:
26176:
26171:
26170:
26160:
26155:
26154:
26150:
26145:
26142:
26141:
26118:
26114:
26103:
26098:
26097:
26093:
26089:
26087:
26085:
26082:
26081:
26074:
26054:
26041:
26036:
26035:
26031:
26030:
26025:
26014:
26009:
26007:
26004:
26003:
25982:
25977:
25976:
25971:
25960:
25955:
25953:
25950:
25949:
25926:
25922:
25916:
25905:
25900:
25896:
25894:
25892:
25889:
25888:
25880:
25874:
25868:
25848:
25835:
25830:
25829:
25825:
25824:
25822:
25819:
25818:
25797:
25792:
25791:
25789:
25786:
25785:
25762:
25758:
25752:
25747:
25739:
25729:
25727:
25725:
25722:
25721:
25714:
25708:
25688:
25675:
25670:
25669:
25665:
25664:
25659:
25654:
25649:
25629:
25626:
25625:
25604:
25599:
25598:
25593:
25588:
25583:
25574:
25570:
25556:
25553:
25552:
25529:
25525:
25519:
25514:
25509:
25505:
25503:
25501:
25498:
25497:
25474:
25470:
25462:
25448:
25445:
25444:
25424:
25413:
25410:
25409:
25386:
25382:
25370:
25348:
25346:
25344:
25341:
25340:
25317:
25306:
25305:
25301:
25297:
25296:
25294:
25291:
25290:
25271:
25270:
25266:
25264:
25261:
25260:
25237:
25233:
25221:
25220:
25216:
25212:
25202:
25200:
25198:
25195:
25194:
25171:
25148:
25143:
25142:
25137:
25131:
25126:
25125:
25124:
25120:
25119:
25107:
25096:
25090:
25087:
25086:
25056:
25051:
25050:
25045:
25039:
25034:
25033:
25021:
25010:
25004:
25001:
25000:
24977:
24973:
24961:
24956:
24955:
24950:
24949:
24945:
24935:
24933:
24931:
24928:
24927:
24920:
24919:
24913:
24907:
24887:
24871:
24866:
24865:
24861:
24860:
24855:
24852:
24851:
24827:
24822:
24821:
24816:
24813:
24812:
24789:
24785:
24774:
24769:
24768:
24764:
24754:
24752:
24750:
24747:
24746:
24739:
24711:
24701:
24697:
24688:
24684:
24678:
24674:
24673:
24671:
24668:
24667:
24648:
24644:
24638:
24634:
24628:
24624:
24618:
24614:
24613:
24597:
24593:
24589:
24587:
24584:
24583:
24560:
24556:
24541:
24537:
24527:
24523:
24513:
24511:
24509:
24506:
24505:
24498:
24492:
24486:
24480:
24459:
24455:
24449:
24445:
24444:
24442:
24439:
24438:
24418:
24416:
24413:
24412:
24389:
24385:
24370:
24357:
24355:
24344:
24340:
24325:
24312:
24310:
24308:
24305:
24304:
24297:
24291:
24285:
24265:
24252:
24247:
24246:
24242:
24241:
24235:
24230:
24229:
24223:
24210:
24205:
24204:
24200:
24199:
24194:
24191:
24190:
24169:
24164:
24163:
24158:
24149:
24144:
24143:
24138:
24135:
24134:
24111:
24107:
24090:
24086:
24085:
24072:
24070:
24068:
24065:
24064:
24056:
24050:
24031:
24020:
24015:
24014:
24006:
24005:
24001:
23999:
23996:
23995:
23972:
23967:
23966:
23958:
23957:
23953:
23947:
23942:
23941:
23939:
23936:
23935:
23912:
23908:
23890:
23886:
23885:
23881:
23871:
23869:
23867:
23864:
23863:
23855:
23849:
23830:
23828:
23825:
23824:
23806:
23801:
23800:
23798:
23795:
23794:
23771:
23767:
23752:
23748:
23747:
23743:
23733:
23731:
23720:
23716:
23704:
23700:
23696:
23692:
23682:
23680:
23678:
23675:
23674:
23666:
23660:
23640:
23635:
23634:
23632:
23629:
23628:
23611:
23609:
23606:
23605:
23582:
23578:
23566:
23553:
23551:
23540:
23536:
23524:
23511:
23509:
23507:
23504:
23503:
23495:
23489:
23469:
23456:
23444:
23443:
23442:
23438:
23437:
23435:
23432:
23431:
23411:
23399:
23398:
23396:
23393:
23392:
23369:
23365:
23347:
23334:
23332:
23330:
23327:
23326:
23316:
23310:
23302:
23296:
23285:
23280:, etc. using a
23273:
23265:
23257:
23251:
23236:
23214:
23210:
23201:
23188:
23186:
23181:
23178:
23177:
23154:
23150:
23141:
23125:
23123:
23121:
23118:
23117:
23101:
23100:
23094:
23088:
23066:
23062:
23053:
23040:
23038:
23027:
23023:
23014:
23001:
22999:
22997:
22994:
22993:
22970:
22966:
22957:
22949:
22936:
22934:
22932:
22929:
22928:
22912:
22897:
22878:
22876:
22873:
22872:
22849:
22845:
22836:
22823:
22821:
22819:
22816:
22815:
22793:
22788:
22787:
22782:
22777:
22771:
22766:
22765:
22756:
22751:
22750:
22745:
22740:
22735:
22733:
22730:
22729:
22711:
22700:
22695:
22694:
22689:
22684:
22678:
22673:
22672:
22663:
22658:
22657:
22652:
22647:
22642:
22641:
22637:
22636:
22634:
22631:
22630:
22607:
22603:
22594:
22589:
22581:
22575:
22571:
22566:
22561:
22554:
22552:
22550:
22547:
22546:
22539:
22533:
22527:
22521:
22501:
22496:
22495:
22487:
22479:
22474:
22460:
22455:
22454:
22446:
22445:
22441:
22439:
22436:
22435:
22417:
22406:
22401:
22400:
22392:
22384:
22379:
22365:
22360:
22359:
22351:
22350:
22346:
22345:
22341:
22340:
22338:
22335:
22334:
22311:
22307:
22298:
22290:
22285:
22277:
22271:
22267:
22262:
22254:
22249:
22242:
22240:
22238:
22235:
22234:
22227:
22221:
22215:
22209:
22189:
22184:
22183:
22178:
22176:
22173:
22172:
22154:
22149:
22148:
22143:
22141:
22138:
22137:
22114:
22110:
22104:
22098:
22093:
22092:
22086:
22081:
22080:
22076:
22074:
22072:
22069:
22068:
22061:
22055:
22049:
22029:
22024:
22023:
22018:
22016:
22013:
22012:
21994:
21989:
21988:
21983:
21981:
21978:
21977:
21954:
21950:
21944:
21939:
21933:
21928:
21927:
21923:
21921:
21919:
21916:
21915:
21908:
21902:
21896:
21890:is being used.
21885:
21882:
21857:
21853:
21849:
21843:
21839:
21837:
21835:
21832:
21831:
21795:
21791:
21787:
21781:
21777:
21775:
21769:
21757:
21753:
21744:
21734:
21732:
21728:
21727:
21726:
21722:
21714:
21711:
21710:
21681:
21677:
21673:
21667:
21663:
21661:
21653:
21649:
21640:
21630:
21628:
21627:
21623:
21615:
21612:
21611:
21584:
21580:
21576:
21567:
21557:
21555:
21553:
21550:
21549:
21532:
21510:
21506:
21498:
21496:
21486:
21469:
21467:
21457:
21440:
21438:
21436:
21433:
21432:
21409:
21405:
21379:
21377:
21375:
21372:
21371:
21355:
21333:
21329:
21321:
21319:
21309:
21292:
21290:
21288:
21285:
21284:
21261:
21257:
21240:
21238:
21236:
21233:
21232:
21216:
21194:
21190:
21182:
21180:
21166:
21162:
21154:
21152:
21147:
21144:
21143:
21120:
21116:
21105:
21103:
21101:
21098:
21097:
21081:
21066:
21044:
21040:
21032:
21030:
21019:
21015:
21007:
21005:
21003:
21000:
20999:
20976:
20972:
20952:
20950:
20948:
20945:
20944:
20928:
20913:
20891:
20887:
20879:
20877:
20872:
20869:
20868:
20845:
20841:
20830:
20828:
20826:
20823:
20822:
20806:
20800:
20794:
20775:
20773:
20770:
20769:
20751:
20746:
20745:
20743:
20740:
20739:
20716:
20712:
20704:
20702:
20700:
20697:
20696:
20689:
20683:
20674:
20666:
20637:
20633:
20625:
20623:
20621:
20618:
20617:
20582:
20578:
20569:
20565:
20559:
20555:
20554:
20537:
20526:
20522:
20507:
20503:
20493:
20489:
20479:
20477:
20475:
20472:
20471:
20442:
20438:
20434:
20424:
20420:
20414:
20410:
20404:
20400:
20394:
20390:
20389:
20378:
20374:
20359:
20355:
20345:
20341:
20331:
20329:
20327:
20324:
20323:
20304:
20303:
20293:
20284:
20272:
20268:
20259:
20255:
20249:
20245:
20244:
20227:
20226:
20222:
20221:
20220:
20216:
20201:
20200:
20190:
20174:
20167:
20163:
20159:
20149:
20145:
20139:
20135:
20129:
20125:
20119:
20115:
20114:
20113:
20109:
20108:
20104:
20089:
20088:
20075:
20061:
20054:
20050:
20046:
20045:
20041:
20019:
20006:
20002:
19996:
19992:
19986:
19982:
19976:
19972:
19971:
19970:
19966:
19951:
19950:
19938:
19934:
19930:
19922:
19908:
19907:
19903:
19882:
19878:
19872:
19868:
19862:
19858:
19852:
19848:
19847:
19839:
19832:
19828:
19813:
19812:
19800:
19796:
19792:
19784:
19770:
19769:
19765:
19746:
19735:
19731:
19726:
19706:
19705:
19701:
19700:
19696:
19681:
19680:
19668:
19664:
19660:
19652:
19648:
19644:
19636:
19613:
19609:
19604:
19584:
19583:
19579:
19564:
19563:
19551:
19547:
19543:
19535:
19521:
19520:
19516:
19491:
19487:
19486:
19482:
19465:
19464:
19460:
19445:
19444:
19432:
19428:
19424:
19410:
19403:
19399:
19374:
19370:
19366:
19362:
19348:
19347:
19343:
19328:
19327:
19315:
19311:
19307:
19293:
19292:
19288:
19270:
19266:
19262:
19245:
19244:
19240:
19225:
19224:
19208:
19204:
19191:
19187:
19183:
19179:
19159:
19155:
19142:
19138:
19122:
19109:
19105:
19095:
19079:
19077:
19074:
19073:
19052:
19048:
19034:
19030:
19020:
19007:
19005:
19003:
19000:
18999:
18982:
18981:
18964:
18935:
18906:
18890:
18873:
18861:
18860:
18849:
18845:
18844:
18840:
18827:
18819:
18806:
18804:
18801:
18800:
18781:
18757:
18753:
18744:
18736:
18732:
18730:
18728:
18725:
18724:
18705:
18701:
18692:
18684:
18680:
18678:
18676:
18673:
18672:
18645:
18640:
18639:
18626:
18620:
18616:
18614:
18597:
18591:
18587:
18585:
18581:
18572:
18567:
18566:
18564:
18561:
18560:
18538:
18532:
18528:
18526:
18518:
18510:
18497:
18491:
18487:
18485:
18483:
18480:
18479:
18454:
18445:
18437:
18430:
18428:
18426:
18423:
18422:
18406:
18391:
18364:
18360:
18351:
18343:
18339:
18337:
18329:
18325:
18316:
18308:
18304:
18302:
18292:
18286:
18282:
18280:
18278:
18275:
18274:
18252:
18246:
18242:
18240:
18232:
18228:
18219:
18211:
18207:
18205:
18197:
18193:
18184:
18176:
18172:
18170:
18168:
18165:
18164:
18139:
18127:
18119:
18111:
18107:
18105:
18103:
18100:
18099:
18083:
18056:
18052:
18043:
18035:
18031:
18029:
18019:
18013:
18009:
18007:
18005:
18002:
18001:
17979:
17973:
17969:
17967:
17959:
17955:
17946:
17938:
17934:
17932:
17930:
17927:
17926:
17901:
17892:
17884:
17880:
17878:
17876:
17873:
17872:
17856:
17836:
17822:
17816:
17812:
17810:
17806:
17805:
17796:
17791:
17790:
17782:
17769:
17763:
17759:
17757:
17755:
17752:
17751:
17729:
17723:
17719:
17717:
17708:
17703:
17702:
17694:
17685:
17671:
17665:
17661:
17659:
17655:
17654:
17652:
17649:
17648:
17623:
17614:
17605:
17600:
17599:
17592:
17590:
17588:
17585:
17584:
17568:
17553:
17529:
17523:
17519:
17517:
17504:
17498:
17494:
17492:
17490:
17487:
17486:
17461:
17452:
17444:
17437:
17435:
17433:
17430:
17429:
17413:
17398:
17378:
17364:
17358:
17354:
17352:
17348:
17347:
17345:
17342:
17341:
17316:
17309:
17304:
17303:
17299:
17297:
17295:
17292:
17291:
17275:
17255:
17250:
17249:
17239:
17233:
17229:
17227:
17225:
17222:
17221:
17199:
17193:
17189:
17187:
17182:
17180:
17177:
17176:
17151:
17145:
17140:
17136:
17134:
17132:
17129:
17128:
17112:
17111:
17101:
17077:
17071:
17067:
17065:
17060:
17057:
17056:
17031:
17025:
17018:
17016:
17014:
17011:
17010:
16994:
16993:
16987:
16981:
16962:
16960:
16957:
16956:
16931:
16925:
16921:
16919:
16917:
16914:
16913:
16906:
16900:
16894:
16888:
16883:
16877:
16846:
16840:
16836:
16834:
16832:
16829:
16828:
16822:
16791:
16783:
16779:
16773:
16765:
16764:
16762:
16760:
16757:
16756:
16733:
16725:
16721:
16714:
16710:
16705:
16697:
16693:
16691:
16689:
16686:
16685:
16662:
16657:
16648:
16640:
16632:
16630:
16628:
16625:
16624:
16617:
16611:
16588:
16587:
16578:
16577:
16571:
16570:
16561:
16560:
16554:
16548:
16547:
16546:
16534:
16533:
16524:
16523:
16517:
16516:
16507:
16501:
16500:
16499:
16493:
16487:
16486:
16485:
16483:
16480:
16479:
16461:
16460:
16454:
16453:
16447:
16443:
16437:
16436:
16427:
16426:
16420:
16419:
16407:
16406:
16400:
16394:
16393:
16392:
16386:
16382:
16376:
16375:
16366:
16365:
16359:
16358:
16353:
16350:
16349:
16325:
16324:
16319:
16309:
16308:
16299:
16298:
16292:
16291:
16282:
16281:
16275:
16271:
16265:
16264:
16255:
16254:
16248:
16247:
16239:
16237:
16235:
16232:
16231:
16224:
16218:
16212:
16206:
16200:
16194:
16174:
16173:
16162:
16156:
16155:
16154:
16148:
16147:
16138:
16132:
16131:
16130:
16124:
16123:
16122:
16118:
16116:
16113:
16112:
16089:
16083:
16082:
16081:
16075:
16074:
16065:
16059:
16058:
16057:
16051:
16050:
16049:
16045:
16039:
16033:
16032:
16031:
16029:
16026:
16025:
16001:
16000:
15995:
15988:
15987:
15981:
15975:
15974:
15973:
15967:
15966:
15960:
15954:
15953:
15952:
15947:
15945:
15943:
15940:
15939:
15932:
15926:
15920:
15896:
15892:
15886:
15882:
15880:
15878:
15875:
15874:
15858:
15850:
15846:
15840:
15836:
15834:
15832:
15829:
15828:
15806:
15802:
15796:
15792:
15790:
15788:
15785:
15784:
15765:
15761:
15755:
15751:
15749:
15743:
15738:
15737:
15735:
15732:
15731:
15708:
15704:
15698:
15692:
15687:
15686:
15682:
15680:
15669:
15665:
15656:
15648:
15641:
15639:
15637:
15634:
15633:
15617:
15616:
15610:
15604:
15585:
15580:
15577:
15576:
15558:
15553:
15552:
15547:
15544:
15543:
15520:
15516:
15509:
15500:
15496:
15495:
15491:
15489:
15478:
15474:
15468:
15462:
15457:
15456:
15452:
15450:
15439:
15435:
15426:
15418:
15411:
15409:
15407:
15404:
15403:
15382:
15377:
15374:
15373:
15349:
15344:
15343:
15335:
15331:
15325:
15320:
15314:
15309:
15308:
15302:
15298:
15297:
15295:
15293:
15290:
15289:
15278:
15277:
15272:
15266:
15246:
15241:
15240:
15232:
15230:
15227:
15226:
15202:
15197:
15196:
15188:
15184:
15178:
15173:
15167:
15162:
15161:
15155:
15151:
15150:
15148:
15146:
15143:
15142:
15135:
15129:
15110:
15105:
15100:
15097:
15096:
15079:
15073:
15068:
15067:
15062:
15059:
15058:
15035:
15031:
15025:
15020:
15014:
15009:
15008:
15004:
15002:
15000:
14997:
14996:
14985:
14984:
14979:
14973:
14954:
14943:
14938:
14937:
14929:
14928:
14924:
14922:
14919:
14918:
14895:
14890:
14889:
14881:
14880:
14876:
14870:
14865:
14864:
14862:
14859:
14858:
14835:
14831:
14825:
14820:
14814:
14809:
14808:
14804:
14802:
14800:
14797:
14796:
14789:
14783:
14764:
14758:
14753:
14752:
14750:
14747:
14746:
14729:
14723:
14718:
14717:
14715:
14712:
14711:
14688:
14684:
14678:
14673:
14667:
14662:
14661:
14657:
14655:
14653:
14650:
14649:
14642:
14636:
14635:
14630:
14624:
14605:
14603:
14600:
14599:
14581:
14576:
14575:
14573:
14570:
14569:
14546:
14542:
14536:
14530:
14525:
14524:
14520:
14518:
14507:
14503:
14497:
14491:
14486:
14485:
14481:
14479:
14477:
14474:
14473:
14472:
14471:
14450:
14446:
14437:
14429:
14422:
14420:
14409:
14405:
14396:
14388:
14381:
14379:
14377:
14374:
14373:
14366:
14360:
14337:
14335:
14332:
14331:
14313:
14308:
14307:
14305:
14302:
14301:
14277:
14272:
14271:
14263:
14259:
14249:
14245:
14244:
14242:
14240:
14237:
14236:
14210:
14206:
14200:
14196:
14194:
14183:
14179:
14173:
14169:
14167:
14165:
14162:
14161:
14145:
14139:
14134:
14133:
14125:
14121:
14115:
14111:
14109:
14101:
14096:
14088:
14084:
14078:
14074:
14072:
14070:
14067:
14066:
14044:
14040:
14034:
14030:
14028:
14017:
14013:
14007:
14003:
14001:
13999:
13996:
13995:
13976:
13972:
13966:
13962:
13960:
13954:
13949:
13948:
13942:
13937:
13936:
13925:
13921:
13915:
13911:
13909:
13904:
13898:
13893:
13892:
13890:
13887:
13886:
13863:
13859:
13853:
13848:
13842:
13837:
13836:
13832:
13830:
13819:
13815:
13806:
13801:
13793:
13786:
13784:
13782:
13779:
13778:
13771:
13765:
13764:
13749:
13734:
13710:
13706:
13700:
13696:
13694:
13683:
13679:
13673:
13669:
13667:
13665:
13662:
13661:
13645:
13637:
13633:
13627:
13623:
13621:
13613:
13605:
13601:
13595:
13591:
13589:
13587:
13584:
13583:
13561:
13557:
13551:
13547:
13545:
13534:
13530:
13524:
13520:
13518:
13516:
13513:
13512:
13493:
13489:
13483:
13479:
13477:
13471:
13466:
13465:
13454:
13450:
13444:
13440:
13438:
13432:
13427:
13426:
13424:
13421:
13420:
13397:
13393:
13387:
13381:
13376:
13375:
13371:
13369:
13358:
13354:
13345:
13337:
13330:
13328:
13326:
13323:
13322:
13306:
13291:
13269:
13265:
13257:
13255:
13245:
13228:
13226:
13216:
13199:
13197:
13195:
13192:
13191:
13168:
13164:
13138:
13136:
13134:
13131:
13130:
13114:
13092:
13088:
13080:
13078:
13068:
13051:
13049:
13047:
13044:
13043:
13020:
13016:
12999:
12997:
12995:
12992:
12991:
12975:
12953:
12949:
12941:
12939:
12925:
12921:
12913:
12911:
12906:
12903:
12902:
12879:
12875:
12864:
12862:
12860:
12857:
12856:
12840:
12825:
12803:
12799:
12791:
12789:
12778:
12774:
12766:
12764:
12762:
12759:
12758:
12735:
12731:
12711:
12709:
12707:
12704:
12703:
12687:
12672:
12650:
12646:
12638:
12636:
12631:
12628:
12627:
12604:
12600:
12589:
12587:
12585:
12582:
12581:
12565:
12564:
12558:
12552:
12533:
12531:
12528:
12527:
12509:
12504:
12503:
12501:
12498:
12497:
12474:
12470:
12462:
12460:
12458:
12455:
12454:
12447:
12441:
12431:
12429:
12420:
12418:
12387:
12386:
12382:
12371:
12367:
12359:
12357:
12355:
12352:
12351:
12342:
12314:
12310:
12301:
12293:
12289:
12287:
12279:
12275:
12266:
12258:
12254:
12252:
12244:
12240:
12234:
12230:
12228:
12226:
12223:
12222:
12202:
12198:
12192:
12188:
12186:
12178:
12174:
12165:
12157:
12153:
12151:
12143:
12139:
12130:
12122:
12118:
12116:
12114:
12111:
12110:
12087:
12083:
12071:
12063:
12055:
12051:
12049:
12047:
12044:
12043:
12027:
12005:
12001:
11992:
11984:
11980:
11978:
11970:
11966:
11960:
11956:
11954:
11952:
11949:
11948:
11928:
11924:
11918:
11914:
11912:
11904:
11900:
11891:
11883:
11879:
11877:
11875:
11872:
11871:
11848:
11844:
11835:
11827:
11823:
11821:
11819:
11816:
11815:
11799:
11777:
11773:
11767:
11763:
11761:
11750:
11746:
11740:
11736:
11734:
11732:
11729:
11728:
11705:
11701:
11692:
11684:
11677:
11675:
11673:
11670:
11669:
11653:
11638:
11618:
11613:
11612:
11604:
11600:
11594:
11590:
11588:
11586:
11583:
11582:
11562:
11558:
11552:
11548:
11546:
11541:
11539:
11536:
11535:
11512:
11508:
11502:
11497:
11493:
11491:
11489:
11486:
11485:
11469:
11468:
11462:
11456:
11436:
11431:
11430:
11422:
11418:
11410:
11408:
11397:
11393:
11387:
11383:
11381:
11376:
11373:
11372:
11352:
11348:
11340:
11338:
11333:
11322:
11318:
11312:
11308:
11306:
11301:
11298:
11297:
11274:
11270:
11264:
11257:
11255:
11253:
11250:
11249:
11233:
11218:
11198:
11193:
11192:
11184:
11180:
11172:
11170:
11168:
11165:
11164:
11144:
11140:
11132:
11130:
11125:
11123:
11120:
11119:
11096:
11092:
11086:
11079:
11077:
11075:
11072:
11071:
11064:
11058:
11057:
11042:
11020:
11016:
11010:
11006:
11004:
10999:
10996:
10995:
10972:
10967:
10961:
10954:
10952:
10950:
10947:
10946:
10930:
10929:
10923:
10917:
10898:
10896:
10893:
10892:
10874:
10869:
10868:
10866:
10863:
10862:
10839:
10835:
10829:
10823:
10818:
10817:
10813:
10811:
10809:
10806:
10805:
10798:
10792:
10772:
10767:
10766:
10764:
10761:
10760:
10743:
10741:
10738:
10737:
10714:
10710:
10704:
10699:
10695:
10693:
10691:
10688:
10687:
10680:
10674:
10655:
10653:
10650:
10649:
10626:
10622:
10616:
10612:
10610:
10608:
10605:
10604:
10583:
10581:
10578:
10577:
10554:
10550:
10544:
10540:
10538:
10536:
10533:
10532:
10525:
10519:
10510:
10504:
10496:
10490:
10461:
10457:
10451:
10447:
10445:
10443:
10440:
10439:
10429:
10417:
10411:
10407:
10401:
10395:
10389:
10383:
10377:
10371:
10365:
10356:
10350:
10346:
10340:
10334:
10328:
10322:
10316:
10310:
10304:
10295:
10289:
10285:
10279:
10273:
10267:
10261:
10255:
10249:
10243:
10210:
10191:
10190:
10180:
10179:
10167:
10163:
10159:
10151:
10149:
10147:
10142:
10130:
10126:
10122:
10114:
10112:
10110:
10098:
10094:
10090:
10082:
10080:
10077:
10076:
10071:
10066:
10061:
10055:
10054:
10042:
10038:
10034:
10026:
10024:
10022:
10017:
10008:
10004:
10000:
9992:
9990:
9988:
9979:
9975:
9971:
9963:
9961:
9958:
9957:
9945:
9941:
9937:
9929:
9927:
9925:
9920:
9911:
9907:
9903:
9895:
9893:
9891:
9882:
9878:
9874:
9866:
9864:
9857:
9856:
9849:
9841:
9837:
9829:
9827:
9824:
9823:
9813:
9812:
9803:
9799:
9795:
9788:
9784:
9780:
9778:
9776:
9771:
9762:
9758:
9754:
9747:
9743:
9739:
9737:
9735:
9726:
9722:
9718:
9711:
9707:
9703:
9701:
9698:
9697:
9692:
9687:
9682:
9676:
9675:
9666:
9662:
9658:
9651:
9647:
9643:
9641:
9639:
9634:
9625:
9621:
9617:
9610:
9606:
9602:
9600:
9598:
9589:
9585:
9581:
9574:
9570:
9566:
9564:
9561:
9560:
9551:
9547:
9543:
9536:
9532:
9528:
9526:
9524:
9519:
9510:
9506:
9502:
9495:
9491:
9487:
9485:
9483:
9474:
9470:
9466:
9459:
9455:
9451:
9449:
9442:
9441:
9434:
9426:
9422:
9416:
9412:
9410:
9407:
9406:
9396:
9395:
9385:
9378:
9374:
9370:
9368:
9366:
9361:
9351:
9344:
9340:
9336:
9334:
9332:
9322:
9315:
9311:
9307:
9305:
9298:
9297:
9290:
9280:
9274:
9270:
9268:
9265:
9264:
9254:
9253:
9244:
9240:
9236:
9228:
9226:
9223:
9222:
9216:
9215:
9206:
9202:
9198:
9190:
9188:
9185:
9184:
9175:
9171:
9167:
9159:
9157:
9150:
9149:
9142:
9134:
9130:
9122:
9120:
9116:
9114:
9111:
9110:
9104:
9085:
9084:
9074:
9073:
9064:
9060:
9055:
9050:
9041:
9037:
9032:
9023:
9019:
9013:
9012:
9007:
9002:
8997:
8991:
8990:
8981:
8977:
8972:
8967:
8961:
8957:
8952:
8946:
8942:
8936:
8935:
8926:
8922:
8917:
8912:
8906:
8902:
8897:
8891:
8887:
8877:
8876:
8869:
8864:
8858:
8857:
8847:
8846:
8836:
8826:
8822:
8818:
8816:
8814:
8809:
8799:
8789:
8785:
8781:
8779:
8777:
8767:
8757:
8753:
8749:
8747:
8744:
8743:
8738:
8733:
8728:
8722:
8721:
8711:
8701:
8697:
8693:
8691:
8689:
8684:
8674:
8667:
8663:
8659:
8657:
8655:
8645:
8638:
8634:
8630:
8628:
8625:
8624:
8614:
8604:
8600:
8596:
8594:
8592:
8587:
8577:
8570:
8566:
8562:
8560:
8558:
8548:
8541:
8537:
8533:
8531:
8524:
8523:
8516:
8506:
8500:
8496:
8494:
8490:
8488:
8485:
8484:
8464:
8463:
8453:
8452:
8440:
8436:
8432:
8424:
8422:
8420:
8415:
8403:
8399:
8395:
8387:
8385:
8383:
8371:
8367:
8363:
8355:
8353:
8350:
8349:
8344:
8339:
8334:
8328:
8327:
8315:
8311:
8307:
8299:
8297:
8295:
8290:
8281:
8277:
8273:
8265:
8263:
8261:
8252:
8248:
8244:
8236:
8234:
8231:
8230:
8218:
8214:
8210:
8202:
8200:
8198:
8193:
8184:
8180:
8176:
8168:
8166:
8164:
8155:
8151:
8147:
8139:
8137:
8130:
8129:
8122:
8114:
8110:
8102:
8100:
8097:
8096:
8086:
8085:
8076:
8072:
8068:
8061:
8057:
8053:
8051:
8049:
8044:
8035:
8031:
8027:
8020:
8016:
8012:
8010:
8008:
7999:
7995:
7991:
7984:
7980:
7976:
7974:
7971:
7970:
7965:
7960:
7955:
7949:
7948:
7939:
7935:
7931:
7924:
7920:
7916:
7914:
7912:
7907:
7898:
7894:
7890:
7883:
7879:
7875:
7873:
7871:
7862:
7858:
7854:
7847:
7843:
7839:
7837:
7834:
7833:
7824:
7820:
7816:
7809:
7805:
7801:
7799:
7797:
7792:
7783:
7779:
7775:
7768:
7764:
7760:
7758:
7756:
7747:
7743:
7739:
7732:
7728:
7724:
7722:
7715:
7714:
7707:
7699:
7695:
7689:
7685:
7683:
7680:
7679:
7669:
7668:
7658:
7651:
7647:
7643:
7641:
7638:
7637:
7631:
7630:
7620:
7613:
7609:
7605:
7603:
7600:
7599:
7589:
7582:
7578:
7574:
7572:
7565:
7564:
7557:
7547:
7541:
7537:
7535:
7532:
7531:
7521:
7520:
7511:
7507:
7503:
7495:
7493:
7491:
7486:
7477:
7473:
7469:
7461:
7459:
7457:
7448:
7444:
7440:
7432:
7430:
7423:
7422:
7415:
7407:
7403:
7395:
7393:
7389:
7387:
7384:
7383:
7377:
7354:
7327:
7323:
7317:
7313:
7311:
7309:
7306:
7305:
7283:
7279:
7273:
7269:
7267:
7265:
7262:
7261:
7249:
7229:
7225:
7217:
7215:
7213:
7210:
7209:
7194:
7192:
7187:
7172:
7164:
7136:
7132:
7126:
7122:
7120:
7118:
7115:
7114:
7102:
7082:
7078:
7072:
7068:
7066:
7064:
7061:
7060:
7052:
7031:
7027:
7019:
7017:
7015:
7012:
7011:
6996:
6994:
6989:
6958:
6936:
6930:
6926:
6924:
6922:
6919:
6918:
6910:
6887:
6881:
6877:
6875:
6873:
6870:
6869:
6844:
6836:
6834:
6832:
6829:
6828:
6818:
6783:
6777:
6767:
6761:
6753:
6690:
6684:
6680:
6678:
6676:
6673:
6672:
6653:
6649:
6641:
6639:
6628:
6624:
6618:
6614:
6612:
6599:
6593:
6589:
6587:
6576:
6572:
6564:
6562:
6560:
6557:
6556:
6526:
6525:
6521:
6513:
6511:
6509:
6506:
6505:
6496:
6477:
6473:
6465:
6463:
6461:
6458:
6457:
6451:
6430:
6424:
6420:
6418:
6416:
6413:
6412:
6400:
6381:
6377:
6369:
6367:
6365:
6362:
6361:
6355:
6334:
6328:
6324:
6322:
6320:
6317:
6316:
6303:
6279:
6273:
6269:
6267:
6265:
6262:
6261:
6255:
6249:
6243:
6237:
6213:
6207:
6203:
6201:
6199:
6196:
6195:
6176:
6172:
6164:
6162:
6160:
6157:
6156:
6132:
6128:
6122:
6118:
6116:
6114:
6111:
6110:
6081:
6075:
6071:
6069:
6067:
6064:
6063:
6040:
6039:
6035:
6027:
6025:
6023:
6020:
6019:
5995:
5989:
5985:
5983:
5981:
5978:
5977:
5958:
5954:
5946:
5944:
5942:
5939:
5938:
5913:
5909:
5903:
5899:
5897:
5895:
5892:
5891:
5867:
5861:
5857:
5855:
5853:
5850:
5849:
5830:
5826:
5818:
5816:
5814:
5811:
5810:
5785:
5781:
5775:
5771:
5769:
5767:
5764:
5763:
5735:
5729:
5725:
5723:
5721:
5718:
5717:
5698:
5694:
5686:
5684:
5682:
5679:
5678:
5665:
5639:
5638:
5634:
5628:
5624:
5622:
5620:
5617:
5616:
5607:
5588:
5584:
5578:
5574:
5572:
5570:
5567:
5566:
5540:
5536:
5528:
5526:
5524:
5521:
5520:
5492:
5486:
5474:
5455:
5451:
5445:
5441:
5439:
5437:
5434:
5433:
5417:
5415:
5412:
5411:
5392:
5388:
5382:
5378:
5376:
5374:
5371:
5370:
5360:
5350:
5344:
5338:
5325:
5319:
5313:
5307:
5297:
5287:
5277:
5271:
5265:
5246:
5242:
5236:
5232:
5230:
5228:
5225:
5224:
5217:
5208:
5164:
5156:
5152:
5144:
5142:
5141:
5137:
5118:
5117:
5113:
5111:
5108:
5107:
5098:
5092:
5081:
5074:vector calculus
5051:
5047:
5038:
5028:
5026:
5015:
5002:
5001:
4997:
4995:
4992:
4991:
4985:vector calculus
4983:In analog with
4949:
4948:
4936:
4932:
4928:
4920:
4918:
4916:
4911:
4899:
4895:
4891:
4883:
4881:
4879:
4867:
4863:
4859:
4851:
4849:
4846:
4845:
4840:
4835:
4830:
4824:
4823:
4811:
4807:
4803:
4795:
4793:
4791:
4786:
4777:
4773:
4769:
4761:
4759:
4757:
4748:
4744:
4740:
4732:
4730:
4727:
4726:
4714:
4710:
4706:
4698:
4696:
4694:
4689:
4680:
4676:
4672:
4664:
4662:
4660:
4651:
4647:
4643:
4635:
4633:
4626:
4625:
4614:
4610:
4602:
4600:
4598:
4595:
4594:
4581:
4571:
4565:
4562:
4536:
4535:
4525:
4515:
4511:
4507:
4505:
4503:
4498:
4488:
4478:
4474:
4470:
4468:
4466:
4456:
4446:
4442:
4438:
4436:
4433:
4432:
4427:
4422:
4417:
4411:
4410:
4400:
4390:
4386:
4382:
4380:
4378:
4373:
4363:
4356:
4352:
4348:
4346:
4344:
4334:
4327:
4323:
4319:
4317:
4314:
4313:
4303:
4293:
4289:
4285:
4283:
4281:
4276:
4266:
4259:
4255:
4251:
4249:
4247:
4237:
4230:
4226:
4222:
4220:
4213:
4212:
4199:
4193:
4189:
4187:
4185:
4182:
4181:
4164:
4158:
4155:
4133:gradient matrix
4124:
4101:
4090:
4086:
4080:
4076:
4074:
4063:
4055:
4050:
4047:
4046:
4040:
4034:
4028:
4021:Jacobian matrix
4007:
4001:
3998:vector calculus
3973:
3972:
3963:
3959:
3955:
3948:
3944:
3940:
3938:
3936:
3931:
3922:
3918:
3914:
3907:
3903:
3899:
3897:
3895:
3886:
3882:
3878:
3871:
3867:
3863:
3861:
3858:
3857:
3852:
3847:
3842:
3836:
3835:
3826:
3822:
3818:
3811:
3807:
3803:
3801:
3799:
3794:
3785:
3781:
3777:
3770:
3766:
3762:
3760:
3758:
3749:
3745:
3741:
3734:
3730:
3726:
3724:
3721:
3720:
3711:
3707:
3703:
3696:
3692:
3688:
3686:
3684:
3679:
3670:
3666:
3662:
3655:
3651:
3647:
3645:
3643:
3634:
3630:
3626:
3619:
3615:
3611:
3609:
3602:
3601:
3590:
3586:
3580:
3576:
3574:
3572:
3569:
3568:
3543:
3542:
3535:
3534:
3528:
3524:
3522:
3517:
3511:
3507:
3505:
3499:
3495:
3488:
3487:
3486:
3478:
3476:
3473:
3472:
3454:
3453:
3446:
3445:
3439:
3435:
3433:
3428:
3422:
3418:
3416:
3410:
3406:
3399:
3398:
3397:
3389:
3387:
3384:
3383:
3380:vector function
3372:
3344:
3336:
3332:
3324:
3322:
3309:
3308:
3304:
3302:
3299:
3298:
3279:
3268:
3248:
3240:
3233:
3232:
3228:
3226:
3223:
3222:
3213:
3207:
3196:
3156:
3155:
3143:
3139:
3131:
3129:
3125:
3124:
3114:
3113:
3104:
3100:
3096:
3088:
3086:
3083:
3082:
3076:
3075:
3066:
3062:
3058:
3050:
3048:
3041:
3040:
3029:
3026:
3025:
3016:
3002:
2995:vector calculus
2970:
2969:
2959:
2955:
2951:
2943:
2940:
2938:
2933:
2923:
2919:
2915:
2907:
2904:
2902:
2892:
2888:
2884:
2876:
2873:
2866:
2865:
2854:
2850:
2842:
2840:
2838:
2835:
2834:
2809:
2808:
2801:
2800:
2794:
2790:
2788:
2783:
2777:
2773:
2771:
2765:
2761:
2754:
2753:
2752:
2744:
2742:
2739:
2738:
2732:
2722:
2704:, which is the
2702:Euclidean space
2677:
2656:
2650:
2646:
2644:
2642:
2639:
2638:
2632:
2620:
2614:
2611:vector calculus
2586:
2585:
2575:
2568:
2564:
2560:
2558:
2555:
2554:
2548:
2547:
2537:
2530:
2526:
2522:
2520:
2517:
2516:
2506:
2499:
2495:
2491:
2489:
2482:
2481:
2468:
2462:
2458:
2456:
2454:
2451:
2450:
2442:is written (in
2437:
2416:
2415:
2408:
2407:
2401:
2397:
2395:
2390:
2384:
2380:
2378:
2372:
2368:
2361:
2360:
2359:
2351:
2349:
2346:
2345:
2335:
2310:
2303:
2297:
2295:Euclidean space
2290:
2279:
2276:vector calculus
2269:
2267:Vector calculus
2263:
2255:differentiation
2242:
2203:
2188:
2186:
2178:
2166:
2161:denotes matrix
2156:
2150:
2144:
2138:
2131:
2125:
2119:
2113:
2098:
2092:
2086:
2080:
2074:
2068:
2058:
2040:
2037:matrix notation
2033:
1999:
1978:
1932:
1928:
1920:
1918:
1916:
1913:
1912:
1885:
1881:
1875:
1871:
1869:
1867:
1864:
1863:
1843:
1839:
1831:
1829:
1827:
1824:
1823:
1796:
1790:
1786:
1784:
1782:
1779:
1778:
1756:
1750:
1746:
1744:
1742:
1739:
1738:
1716:
1708:
1706:
1704:
1701:
1700:
1661:
1657:
1653:
1650:gradient matrix
1625:
1624:
1617:
1616:
1606:
1602:
1598:
1590:
1587:
1585:
1575:
1571:
1567:
1559:
1556:
1554:
1544:
1540:
1536:
1528:
1525:
1518:
1517:
1516:
1506:
1505:
1493:
1489:
1481:
1479:
1475:
1474:
1463:
1460:
1459:
1440:
1438:
1435:
1434:
1402:
1399:
1398:
1381:
1377:
1375:
1372:
1371:
1354:
1343:
1342:
1341:
1339:
1336:
1335:
1312:
1301:
1300:
1299:
1290:
1286:
1282:
1274:
1272:
1263:
1252:
1251:
1250:
1241:
1237:
1233:
1225:
1223:
1214:
1203:
1202:
1201:
1192:
1188:
1184:
1176:
1174:
1163:
1160:
1159:
1136:
1132:
1123:
1119:
1110:
1106:
1098:
1095:
1094:
1091:vector calculus
1075:
992:matrix calculus
980:
951:
950:
936:Integration Bee
911:
908:
901:
900:
876:
873:
866:
865:
838:
835:
828:
827:
809:Volume integral
744:
739:
732:
731:
648:
643:
636:
635:
605:
524:
519:
512:
511:
503:Risch algorithm
478:Euler's formula
364:
359:
352:
351:
333:General Leibniz
216:generalizations
198:
193:
186:
172:Rolle's theorem
167:
142:
78:
72:
67:
61:
58:
57:
39:
36:vector calculus
28:
23:
22:
15:
12:
11:
5:
36869:
36859:
36858:
36853:
36851:Linear algebra
36848:
36831:
36830:
36828:
36827:
36822:
36817:
36812:
36807:
36805:Gabriel's horn
36802:
36797:
36796:
36795:
36790:
36785:
36780:
36775:
36767:
36766:
36765:
36756:
36754:
36750:
36749:
36746:
36745:
36743:
36742:
36737:
36735:List of limits
36731:
36728:
36727:
36725:
36724:
36723:
36722:
36717:
36712:
36702:
36701:
36700:
36690:
36685:
36680:
36675:
36669:
36667:
36658:
36654:
36653:
36651:
36650:
36643:
36636:
36634:Leonhard Euler
36631:
36626:
36621:
36616:
36611:
36606:
36601:
36596:
36591:
36586:
36580:
36578:
36572:
36571:
36569:
36568:
36563:
36558:
36553:
36548:
36542:
36540:
36534:
36533:
36531:
36530:
36529:
36528:
36523:
36518:
36513:
36508:
36503:
36498:
36493:
36488:
36483:
36475:
36474:
36473:
36468:
36467:
36466:
36461:
36451:
36446:
36441:
36436:
36431:
36426:
36418:
36412:
36410:
36406:
36405:
36403:
36402:
36401:
36400:
36395:
36390:
36385:
36377:
36372:
36367:
36362:
36357:
36352:
36347:
36342:
36337:
36335:Hessian matrix
36332:
36327:
36321:
36319:
36313:
36312:
36310:
36309:
36308:
36307:
36302:
36297:
36292:
36290:Line integrals
36284:
36283:
36282:
36277:
36272:
36267:
36262:
36253:
36251:
36245:
36244:
36242:
36241:
36236:
36231:
36230:
36229:
36224:
36216:
36211:
36210:
36209:
36199:
36198:
36197:
36192:
36187:
36177:
36172:
36171:
36170:
36160:
36155:
36150:
36145:
36140:
36138:Antiderivative
36134:
36132:
36126:
36125:
36123:
36122:
36121:
36120:
36115:
36110:
36100:
36099:
36098:
36093:
36085:
36084:
36083:
36078:
36073:
36068:
36058:
36057:
36056:
36051:
36046:
36041:
36033:
36032:
36031:
36026:
36025:
36024:
36014:
36009:
36004:
35999:
35994:
35984:
35983:
35982:
35977:
35967:
35962:
35957:
35952:
35947:
35942:
35936:
35934:
35928:
35927:
35925:
35924:
35919:
35914:
35909:
35908:
35907:
35897:
35891:
35889:
35883:
35882:
35880:
35879:
35874:
35869:
35864:
35859:
35854:
35849:
35844:
35839:
35834:
35829:
35824:
35819:
35813:
35811:
35805:
35804:
35797:
35796:
35789:
35782:
35774:
35768:
35767:
35761:
35755:
35749:
35739:
35722:
35712:
35706:
35694:
35691:
35690:
35689:
35675:
35665:
35657:
35654:
35652:
35651:External links
35649:
35647:
35646:
35624:
35618:
35605:
35591:
35583:Matrix algebra
35577:
35575:
35572:
35569:
35568:
35537:
35504:
35497:
35479:
35472:
35454:
35447:
35433:Pan, Jianxin;
35425:
35418:
35397:
35385:
35370:
35343:
35330:
35304:
35292:
35285:
35281:
35276:
35273:
35244:
35238:
35235:
35229:
35225:
35143:
35100:
35067:
35066:
35064:
35061:
35058:
35057:
35045:
35039:
35036:
35031:
35028:
35025:
35022:
35016:
35010:
35007:
35002:
34999:
34996:
34993:
34987:
34981:
34978:
34973:
34970:
34967:
34964:
34958:
34952:
34949:
34944:
34941:
34938:
34935:
34932:
34929:
34926:
34923:
34920:
34917:
34911:
34905:
34902:
34897:
34894:
34891:
34888:
34885:
34862:
34856:
34853:
34848:
34845:
34842:
34839:
34833:
34827:
34824:
34819:
34816:
34808:
34805:
34800:
34794:
34791:
34786:
34783:
34777:
34771:
34768:
34764:
34759:
34753:
34750:
34745:
34742:
34739:
34736:
34733:
34724:
34721:
34717:
34712:
34706:
34703:
34698:
34695:
34692:
34689:
34686:
34661:
34641:
34621:
34589:
34566:
34565:
34563:
34560:
34559:
34558:
34553:
34551:Ricci calculus
34548:
34543:
34537:
34536:
34520:
34517:
34503:It is used in
34485:
34482:
34481:
34480:
34477:
34476:
34464:
34460:
34454:
34451:
34445:
34441:
34428:
34417:
34414:
34409:
34405:
34401:
34397:
34386:
34385:
34373:
34369:
34362:
34358:
34352:
34348:
34335:
34323:
34319:
34314:
34310:
34306:
34302:
34291:
34290:
34278:
34274:
34268:
34265:
34259:
34255:
34242:
34231:
34228:
34223:
34219:
34215:
34211:
34200:
34199:
34187:
34183:
34176:
34172:
34167:
34164:
34151:
34140:
34136:
34132:
34127:
34123:
34120:
34117:
34114:
34111:
34108:
34097:
34096:
34083:
34078:
34073:
34066:
34062:
34057:
34054:
34041:
34029:
34025:
34020:
34015:
34010:
34007:
34004:
33993:
33992:
33981:
33978:
33972:
33969:
33964:
33961:
33948:
33937:
33934:
33930:
33927:
33924:
33921:
33910:
33909:
33906:
33875:
33870:
33863:
33860:
33856:
33852:
33847:
33842:
33835:
33830:
33805:
33800:
33793:
33789:
33783:
33779:
33775:
33771:
33748:
33743:
33738:
33733:
33729:
33725:
33722:
33717:
33713:
33709:
33706:
33702:
33698:
33695:
33675:
33672:
33669:
33666:
33643:
33639:
33635:
33632:
33610:
33606:
33602:
33597:
33594:
33590:
33585:
33581:
33559:
33556:
33551:
33545:
33540:
33536:
33532:
33489:
33486:
33482:
33476:
33473:
33468:
33463:
33458:
33455:
33451:
33446:
33442:
33439:
33434:
33431:
33427:
33421:
33416:
33411:
33385:
33382:
33378:
33366:
33365:
33362:
33361:
33348:
33341:
33337:
33333:
33328:
33324:
33320:
33313:
33309:
33305:
33300:
33296:
33290:
33285:
33281:
33277:
33274:
33271:
33268:
33263:
33259:
33255:
33252:
33246:
33245:
33240:
33236:
33232:
33227:
33223:
33219:
33217:
33212:
33208:
33204:
33200:
33197:
33193:
33192:
33190:
33183:
33178:
33173:
33169:
33165:
33162:
33157:
33152:
33145:
33142:
33138:
33127:
33116:
33112:
33108:
33104:
33100:
33097:
33093:
33089:
33079:
33065:
33061:
33049:differentiable
33028:
33023:
33016:
33013:
33009:
33005:
33000:
32995:
32988:
32983:
32969:diagonalizable
32954:
32949:
32942:
32938:
32932:
32928:
32924:
32920:
32908:
32907:
32896:
32893:
32887:
32884:
32880:
32875:
32870:
32867:
32863:
32859:
32856:
32852:
32848:
32845:
32840:
32837:
32833:
32828:
32823:
32820:
32816:
32812:
32807:
32802:
32798:
32787:
32776:
32772:
32767:
32763:
32760:
32756:
32752:
32742:
32739:
32738:
32727:
32724:
32717:
32713:
32710:
32707:
32704:
32701:
32697:
32693:
32689:
32685:
32682:
32676:
32672:
32668:
32662:
32657:
32653:
32642:
32631:
32627:
32621:
32616:
32612:
32608:
32598:
32595:
32594:
32581:
32578:
32575:
32572:
32569:
32564:
32559:
32555:
32551:
32548:
32543:
32538:
32531:
32528:
32525:
32520:
32517:
32514:
32510:
32499:
32488:
32484:
32479:
32474:
32469:
32465:
32455:
32448:
32447:
32433:
32428:
32423:
32419:
32416:
32406:
32395:
32391:
32385:
32379:
32374:
32370:
32360:
32352:
32351:
32338:
32335:
32330:
32324:
32319:
32315:
32311:
32305:
32302:
32297:
32292:
32282:
32271:
32267:
32262:
32259:
32254:
32249:
32245:
32235:
32232:
32231:
32218:
32214:
32209:
32205:
32202:
32192:
32181:
32177:
32172:
32167:
32162:
32158:
32148:
32145:
32144:
32133:
32129:
32125:
32122:
32119:
32115:
32111:
32107:
32103:
32100:
32096:
32092:
32089:
32079:
32068:
32065:
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32057:
32053:
32049:
32046:
32036:
32028:
32027:
32016:
32012:
32008:
32005:
32002:
31998:
31994:
31990:
31986:
31983:
31979:
31975:
31972:
31962:
31951:
31948:
31944:
31940:
31936:
31932:
31929:
31919:
31911:
31910:
31899:
31895:
31891:
31888:
31884:
31880:
31876:
31872:
31868:
31864:
31861:
31851:
31840:
31837:
31833:
31828:
31824:
31821:
31811:
31808:
31807:
31795:
31791:
31788:
31784:
31780:
31770:
31759:
31756:
31752:
31748:
31744:
31740:
31737:
31727:
31724:
31723:
31711:
31707:
31703:
31693:
31682:
31679:
31675:
31671:
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31655:
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31642:
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31621:
31610:
31607:
31603:
31599:
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31570:
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31556:
31555:
31552:
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31539:
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31002:
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30907:
30894:
30884:
30870:
30799:
30798:
30786:
30779:
30776:
30770:
30766:
30758:
30753:
30746:
30742:
30737:
30733:
30729:
30726:
30723:
30717:
30711:
30707:
30704:
30694:
30682:
30675:
30672:
30666:
30662:
30652:
30648:
30643:
30639:
30635:
30632:
30629:
30622:
30618:
30615:
30605:
30594:
30588:
30585:
30580:
30576:
30572:
30569:
30566:
30553:
30536:
30535:
30523:
30518:
30513:
30508:
30501:
30498:
30492:
30488:
30480:
30477:
30472:
30466:
30461:
30457:
30454:
30451:
30447:
30440:
30437:
30431:
30427:
30419:
30416:
30411:
30405:
30401:
30396:
30392:
30388:
30384:
30375:
30371:
30367:
30361:
30355:
30351:
30342:
30339:
30334:
30328:
30324:
30321:
30317:
30312:
30308:
30304:
30293:
30282:
30274:
30270:
30266:
30260:
30255:
30250:
30244:
30240:
30226:
30209:
30208:
30196:
30189:
30186:
30180:
30176:
30168:
30165:
30160:
30154:
30150:
30147:
30137:
30126:
30120:
30117:
30111:
30106:
30101:
30097:
30094:
30091:
30078:
30061:
30060:
30048:
30041:
30038:
30032:
30028:
30020:
30017:
30012:
30006:
30002:
29999:
29995:
29990:
29985:
29974:
29963:
29957:
29954:
29948:
29943:
29938:
29934:
29921:
29904:
29903:
29887:
29871:
29868:
29856:
29853:
29852:
29851:
29848:
29847:
29835:
29831:
29825:
29822:
29816:
29812:
29806:
29800:
29797:
29791:
29787:
29781:
29777:
29766:
29755:
29749:
29746:
29741:
29737:
29733:
29729:
29725:
29722:
29709:
29677:
29676:
29662:
29659:
29653:
29649:
29643:
29636:
29632:
29627:
29623:
29619:
29616:
29613:
29600:
29589:
29583:
29580:
29575:
29571:
29567:
29564:
29561:
29548:
29531:
29530:
29523:
29520:
29508:
29505:
29503:
29500:
29493:
29492:
29489:
29488:
29476:
29469:
29465:
29461:
29457:
29451:
29447:
29443:
29438:
29427:
29416:
29410:
29407:
29399:
29395:
29391:
29387:
29374:
29360:
29359:
29347:
29343:
29339:
29335:
29332:
29328:
29324:
29319:
29316:
29312:
29308:
29305:
29301:
29297:
29291:
29280:
29269:
29263:
29260:
29255:
29251:
29247:
29244:
29240:
29235:
29222:
29212:
29198:
29127:
29126:
29113:
29110:
29105:
29099:
29092:
29089:
29083:
29079:
29071:
29068:
29063:
29055:
29052:
29046:
29042:
29036:
29030:
29027:
29024:
29021:
29015:
29009:
29005:
28998:
28992:
28989:
28983:
28979:
28971:
28968:
28963:
28955:
28952:
28946:
28942:
28935:
28929:
28926:
28921:
28909:
28898:
28892:
28889:
28886:
28883:
28876:
28873:
28868:
28861:
28857:
28843:
28822:
28821:
28808:
28805:
28800:
28792:
28789:
28783:
28779:
28771:
28768:
28763:
28758:
28748:
28737:
28731:
28728:
28721:
28718:
28713:
28708:
28695:
28678:
28677:
28665:
28661:
28655:
28652:
28646:
28642:
28636:
28630:
28627:
28621:
28617:
28611:
28607:
28596:
28585:
28579:
28576:
28571:
28567:
28563:
28559:
28555:
28552:
28539:
28507:
28506:
28494:
28490:
28484:
28481:
28475:
28471:
28465:
28459:
28456:
28450:
28446:
28440:
28436:
28425:
28414:
28408:
28405:
28400:
28396:
28392:
28388:
28384:
28381:
28368:
28336:
28335:
28323:
28316:
28313:
28307:
28303:
28297:
28291:
28288:
28282:
28278:
28271:
28260:
28249:
28243:
28240:
28235:
28231:
28226:
28222:
28219:
28206:
28174:
28173:
28159:
28156:
28150:
28146:
28140:
28134:
28131:
28125:
28121:
28108:
28097:
28091:
28088:
28083:
28079:
28075:
28071:
28067:
28064:
28051:
28019:
28018:
28006:
27999:
27996:
27990:
27986:
27979:
27968:
27957:
27951:
27948:
27942:
27939:
27936:
27932:
27919:
27885:
27884:
27870:
27867:
27861:
27857:
27851:
27841:
27830:
27824:
27821:
27815:
27811:
27808:
27795:
27778:
27777:
27769:
27766:
27747:
27744:
27738:
27734:
27716:
27713:
27712:
27711:
27708:
27707:
27690:
27683:
27680:
27675:
27671:
27666:
27662:
27656:
27652:
27647:
27641:
27636:
27632:
27627:
27625:
27620:
27615:
27612:
27607:
27603:
27600:
27595:
27591:
27586:
27580:
27577:
27573:
27569:
27563:
27558:
27553:
27546:
27542:
27536:
27532:
27531:
27521:
27504:
27496:
27492:
27486:
27482:
27475:
27472:
27467:
27463:
27458:
27454:
27448:
27444:
27439:
27434:
27432:
27427:
27423:
27418:
27414:
27407:
27404:
27399:
27395:
27392:
27387:
27383:
27378:
27373:
27369:
27363:
27358:
27353:
27346:
27342:
27336:
27332:
27331:
27321:
27310:
27303:
27299:
27293:
27288:
27283:
27276:
27272:
27266:
27262:
27249:
27241:is non-square,
27220:
27219:
27206:
27203:
27198:
27194:
27191:
27186:
27182:
27177:
27171:
27168:
27163:
27158:
27153:
27146:
27142:
27136:
27132:
27122:
27108:
27104:
27098:
27094:
27087:
27084:
27079:
27075:
27070:
27066:
27060:
27056:
27051:
27045:
27040:
27035:
27028:
27024:
27018:
27014:
27004:
26993:
26986:
26982:
26976:
26971:
26966:
26959:
26955:
26949:
26945:
26932:
26924:is non-square,
26903:
26902:
26889:
26884:
26879:
26876:
26871:
26866:
26860:
26855:
26850:
26843:
26839:
26833:
26829:
26819:
26806:
26803:
26798:
26792:
26787:
26782:
26777:
26772:
26767:
26762:
26756:
26752:
26747:
26743:
26740:
26735:
26732:
26727:
26721:
26716:
26711:
26704:
26700:
26694:
26690:
26680:
26669:
26662:
26658:
26652:
26647:
26642:
26635:
26631:
26625:
26621:
26608:
26586:
26585:
26572:
26567:
26562:
26559:
26549:
26537:
26533:
26530:
26520:
26509:
26501:
26496:
26491:
26485:
26480:
26474:
26469:
26463:
26459:
26456:
26453:
26439:
26436:pseudo-inverse
26431:
26430:
26417:
26412:
26407:
26402:
26397:
26392:
26382:
26369:
26364:
26359:
26349:
26338:
26331:
26327:
26321:
26316:
26310:
26305:
26299:
26295:
26292:
26289:
26275:
26272:pseudo-inverse
26267:
26266:
26253:
26248:
26243:
26240:
26235:
26230:
26224:
26219:
26214:
26209:
26205:
26195:
26182:
26179:
26174:
26168:
26163:
26158:
26153:
26149:
26139:
26128:
26121:
26117:
26111:
26106:
26101:
26096:
26092:
26078:
26071:
26070:
26057:
26052:
26047:
26044:
26039:
26034:
26028:
26023:
26020:
26017:
26012:
26001:
25988:
25985:
25980:
25974:
25969:
25966:
25963:
25958:
25947:
25936:
25929:
25925:
25919:
25914:
25911:
25908:
25903:
25899:
25885:
25865:
25864:
25851:
25846:
25841:
25838:
25833:
25828:
25816:
25803:
25800:
25795:
25783:
25772:
25765:
25761:
25755:
25750:
25746:
25742:
25738:
25735:
25732:
25719:
25705:
25704:
25691:
25686:
25681:
25678:
25673:
25668:
25662:
25657:
25652:
25648:
25645:
25642:
25639:
25636:
25633:
25623:
25610:
25607:
25602:
25596:
25591:
25586:
25582:
25577:
25573:
25569:
25566:
25563:
25560:
25550:
25539:
25532:
25528:
25522:
25517:
25512:
25508:
25494:
25491:
25490:
25477:
25473:
25469:
25465:
25461:
25458:
25455:
25452:
25442:
25431:
25427:
25423:
25420:
25417:
25407:
25396:
25389:
25385:
25380:
25377:
25373:
25369:
25366:
25363:
25360:
25357:
25354:
25351:
25337:
25334:
25333:
25320:
25315:
25309:
25304:
25300:
25288:
25274:
25269:
25258:
25247:
25240:
25236:
25230:
25224:
25219:
25215:
25211:
25208:
25205:
25191:
25188:
25187:
25174:
25169:
25163:
25160:
25157:
25154:
25151:
25146:
25140:
25134:
25129:
25123:
25116:
25113:
25110:
25105:
25102:
25099:
25095:
25084:
25071:
25068:
25065:
25062:
25059:
25054:
25048:
25042:
25037:
25030:
25027:
25024:
25019:
25016:
25013:
25009:
24998:
24987:
24980:
24976:
24970:
24964:
24959:
24953:
24948:
24944:
24941:
24938:
24924:
24904:
24903:
24890:
24885:
24880:
24877:
24874:
24869:
24864:
24859:
24849:
24836:
24833:
24830:
24825:
24820:
24810:
24799:
24792:
24788:
24782:
24777:
24772:
24767:
24763:
24760:
24757:
24743:
24736:
24735:
24723:
24720:
24717:
24714:
24710:
24704:
24700:
24696:
24691:
24687:
24681:
24677:
24665:
24651:
24647:
24641:
24637:
24631:
24627:
24621:
24617:
24612:
24608:
24605:
24600:
24596:
24592:
24581:
24570:
24563:
24559:
24553:
24549:
24544:
24540:
24536:
24533:
24530:
24526:
24522:
24519:
24516:
24503:
24477:
24476:
24462:
24458:
24452:
24448:
24436:
24424:
24421:
24410:
24399:
24392:
24388:
24383:
24379:
24376:
24373:
24369:
24366:
24363:
24360:
24354:
24347:
24343:
24338:
24334:
24331:
24328:
24324:
24321:
24318:
24315:
24302:
24282:
24281:
24268:
24263:
24258:
24255:
24250:
24245:
24238:
24233:
24226:
24221:
24216:
24213:
24208:
24203:
24198:
24188:
24175:
24172:
24167:
24161:
24155:
24152:
24147:
24142:
24132:
24121:
24114:
24110:
24105:
24101:
24096:
24093:
24089:
24084:
24081:
24078:
24075:
24061:
24047:
24046:
24034:
24029:
24023:
24018:
24013:
24009:
24004:
23993:
23981:
23975:
23970:
23965:
23961:
23956:
23950:
23945:
23933:
23922:
23915:
23911:
23905:
23901:
23898:
23893:
23889:
23884:
23880:
23877:
23874:
23860:
23846:
23845:
23833:
23822:
23809:
23804:
23792:
23781:
23774:
23770:
23764:
23760:
23755:
23751:
23746:
23742:
23739:
23736:
23730:
23723:
23719:
23713:
23707:
23703:
23699:
23695:
23691:
23688:
23685:
23671:
23657:
23656:
23643:
23638:
23626:
23614:
23603:
23592:
23585:
23581:
23576:
23572:
23569:
23565:
23562:
23559:
23556:
23550:
23543:
23539:
23534:
23530:
23527:
23523:
23520:
23517:
23514:
23500:
23486:
23485:
23472:
23467:
23463:
23459:
23455:
23451:
23447:
23441:
23429:
23418:
23414:
23410:
23406:
23402:
23390:
23379:
23372:
23368:
23363:
23359:
23356:
23353:
23350:
23346:
23343:
23340:
23337:
23324:
23314:
23300:
23233:
23232:
23217:
23213:
23208:
23204:
23200:
23197:
23194:
23191:
23185:
23175:
23164:
23157:
23153:
23148:
23144:
23140:
23137:
23134:
23131:
23128:
23115:
23085:
23084:
23069:
23065:
23060:
23056:
23052:
23049:
23046:
23043:
23037:
23030:
23026:
23021:
23017:
23013:
23010:
23007:
23004:
22991:
22980:
22973:
22969:
22964:
22960:
22956:
22952:
22948:
22945:
22942:
22939:
22926:
22894:
22893:
22881:
22870:
22859:
22852:
22848:
22843:
22839:
22835:
22832:
22829:
22826:
22813:
22810:
22809:
22796:
22791:
22785:
22780:
22774:
22769:
22764:
22759:
22754:
22748:
22743:
22738:
22727:
22714:
22709:
22703:
22698:
22692:
22687:
22681:
22676:
22671:
22666:
22661:
22655:
22650:
22645:
22640:
22628:
22617:
22610:
22606:
22601:
22597:
22592:
22588:
22584:
22578:
22574:
22569:
22564:
22560:
22557:
22544:
22518:
22517:
22504:
22499:
22494:
22490:
22486:
22482:
22477:
22473:
22469:
22463:
22458:
22453:
22449:
22444:
22433:
22420:
22415:
22409:
22404:
22399:
22395:
22391:
22387:
22382:
22378:
22374:
22368:
22363:
22358:
22354:
22349:
22344:
22332:
22321:
22314:
22310:
22305:
22301:
22297:
22293:
22288:
22284:
22280:
22274:
22270:
22265:
22261:
22257:
22252:
22248:
22245:
22232:
22206:
22205:
22192:
22187:
22181:
22170:
22157:
22152:
22146:
22135:
22124:
22117:
22113:
22107:
22101:
22096:
22089:
22084:
22079:
22066:
22046:
22045:
22032:
22027:
22021:
22010:
21997:
21992:
21986:
21975:
21964:
21957:
21953:
21947:
21942:
21936:
21931:
21926:
21913:
21893:
21892:
21871:
21863:
21860:
21856:
21852:
21846:
21842:
21823:
21822:
21810:
21801:
21798:
21794:
21790:
21784:
21780:
21772:
21767:
21760:
21756:
21751:
21747:
21743:
21740:
21737:
21731:
21725:
21721:
21718:
21708:
21696:
21687:
21684:
21680:
21676:
21670:
21666:
21656:
21652:
21647:
21643:
21639:
21636:
21633:
21626:
21622:
21619:
21609:
21598:
21590:
21587:
21583:
21579:
21574:
21570:
21566:
21563:
21560:
21546:
21529:
21528:
21513:
21509:
21504:
21501:
21492:
21489:
21484:
21481:
21478:
21475:
21472:
21463:
21460:
21455:
21452:
21449:
21446:
21443:
21430:
21419:
21412:
21408:
21403:
21400:
21397:
21394:
21391:
21388:
21385:
21382:
21369:
21352:
21351:
21336:
21332:
21327:
21324:
21315:
21312:
21307:
21304:
21301:
21298:
21295:
21282:
21271:
21264:
21260:
21255:
21252:
21249:
21246:
21243:
21230:
21213:
21212:
21197:
21193:
21188:
21185:
21179:
21176:
21169:
21165:
21160:
21157:
21151:
21141:
21130:
21123:
21119:
21114:
21111:
21108:
21095:
21063:
21062:
21047:
21043:
21038:
21035:
21029:
21022:
21018:
21013:
21010:
20997:
20986:
20979:
20975:
20970:
20967:
20964:
20961:
20958:
20955:
20942:
20910:
20909:
20894:
20890:
20885:
20882:
20876:
20866:
20855:
20848:
20844:
20839:
20836:
20833:
20820:
20791:
20790:
20778:
20767:
20754:
20749:
20737:
20726:
20719:
20715:
20710:
20707:
20694:
20680:
20679:
20671:
20663:
20660:
20640:
20636:
20631:
20628:
20604:
20603:
20591:
20585:
20581:
20577:
20572:
20568:
20562:
20558:
20553:
20549:
20546:
20543:
20540:
20536:
20529:
20525:
20519:
20515:
20510:
20506:
20502:
20499:
20496:
20492:
20488:
20485:
20482:
20469:
20457:
20453:
20450:
20445:
20441:
20437:
20433:
20427:
20423:
20417:
20413:
20407:
20403:
20397:
20393:
20388:
20381:
20377:
20371:
20367:
20362:
20358:
20354:
20351:
20348:
20344:
20340:
20337:
20334:
20301:
20296:
20292:
20287:
20282:
20275:
20271:
20267:
20262:
20258:
20252:
20248:
20243:
20239:
20236:
20233:
20230:
20225:
20219:
20215:
20212:
20209:
20206:
20204:
20202:
20198:
20193:
20189:
20185:
20180:
20177:
20170:
20166:
20162:
20158:
20152:
20148:
20142:
20138:
20132:
20128:
20122:
20118:
20112:
20107:
20103:
20100:
20097:
20094:
20092:
20090:
20086:
20082:
20078:
20074:
20071:
20067:
20064:
20057:
20053:
20049:
20044:
20040:
20037:
20034:
20030:
20026:
20022:
20018:
20015:
20009:
20005:
19999:
19995:
19989:
19985:
19979:
19975:
19969:
19965:
19962:
19959:
19956:
19954:
19952:
19948:
19941:
19937:
19933:
19929:
19925:
19921:
19918:
19914:
19911:
19906:
19902:
19899:
19896:
19892:
19885:
19881:
19875:
19871:
19865:
19861:
19855:
19851:
19846:
19842:
19838:
19835:
19831:
19827:
19824:
19821:
19818:
19816:
19814:
19810:
19803:
19799:
19795:
19791:
19787:
19783:
19780:
19776:
19773:
19768:
19764:
19761:
19758:
19754:
19749:
19744:
19738:
19734:
19729:
19725:
19722:
19718:
19715:
19712:
19709:
19704:
19699:
19695:
19692:
19689:
19686:
19684:
19682:
19678:
19671:
19667:
19663:
19659:
19655:
19651:
19647:
19642:
19639:
19635:
19632:
19629:
19626:
19622:
19616:
19612:
19607:
19603:
19600:
19596:
19593:
19590:
19587:
19582:
19578:
19575:
19572:
19569:
19567:
19565:
19561:
19554:
19550:
19546:
19542:
19538:
19534:
19531:
19527:
19524:
19519:
19515:
19512:
19509:
19505:
19500:
19494:
19490:
19485:
19481:
19477:
19474:
19471:
19468:
19463:
19459:
19456:
19453:
19450:
19448:
19446:
19442:
19435:
19431:
19427:
19423:
19419:
19416:
19413:
19409:
19406:
19402:
19398:
19395:
19392:
19388:
19383:
19377:
19373:
19369:
19365:
19361:
19357:
19354:
19351:
19346:
19342:
19339:
19336:
19333:
19331:
19329:
19325:
19318:
19314:
19310:
19306:
19302:
19299:
19296:
19291:
19287:
19284:
19280:
19273:
19269:
19265:
19261:
19258:
19254:
19251:
19248:
19243:
19239:
19236:
19233:
19230:
19228:
19226:
19222:
19217:
19211:
19207:
19203:
19200:
19197:
19194:
19190:
19186:
19182:
19178:
19175:
19172:
19168:
19162:
19158:
19154:
19151:
19148:
19145:
19141:
19137:
19134:
19131:
19128:
19125:
19123:
19121:
19117:
19112:
19108:
19104:
19101:
19098:
19094:
19091:
19088:
19085:
19082:
19081:
19062:
19055:
19051:
19046:
19042:
19037:
19033:
19029:
19026:
19023:
19019:
19016:
19013:
19010:
18996:
18995:
18980:
18976:
18973:
18970:
18967:
18963:
18960:
18957:
18954:
18951:
18947:
18944:
18941:
18938:
18934:
18931:
18928:
18925:
18922:
18918:
18915:
18912:
18909:
18905:
18902:
18899:
18896:
18893:
18891:
18889:
18885:
18882:
18879:
18876:
18872:
18869:
18866:
18863:
18862:
18858:
18852:
18848:
18843:
18839:
18836:
18833:
18830:
18828:
18826:
18822:
18818:
18815:
18812:
18809:
18808:
18780:
18777:
18760:
18756:
18751:
18747:
18743:
18739:
18735:
18708:
18704:
18699:
18695:
18691:
18687:
18683:
18666:
18665:
18662:
18661:
18648:
18643:
18638:
18632:
18629:
18623:
18619:
18613:
18609:
18603:
18600:
18594:
18590:
18584:
18580:
18575:
18570:
18558:
18544:
18541:
18535:
18531:
18525:
18521:
18517:
18513:
18509:
18503:
18500:
18494:
18490:
18477:
18466:
18460:
18457:
18452:
18448:
18444:
18440:
18436:
18433:
18420:
18388:
18387:
18383:
18382:
18367:
18363:
18358:
18354:
18350:
18346:
18342:
18332:
18328:
18323:
18319:
18315:
18311:
18307:
18298:
18295:
18289:
18285:
18272:
18258:
18255:
18249:
18245:
18235:
18231:
18226:
18222:
18218:
18214:
18210:
18200:
18196:
18191:
18187:
18183:
18179:
18175:
18162:
18151:
18145:
18142:
18137:
18134:
18130:
18126:
18122:
18118:
18114:
18110:
18097:
18080:
18079:
18075:
18074:
18059:
18055:
18050:
18046:
18042:
18038:
18034:
18025:
18022:
18016:
18012:
17999:
17985:
17982:
17976:
17972:
17962:
17958:
17953:
17949:
17945:
17941:
17937:
17924:
17913:
17907:
17904:
17899:
17895:
17891:
17887:
17883:
17870:
17853:
17852:
17839:
17834:
17828:
17825:
17819:
17815:
17809:
17804:
17799:
17794:
17789:
17785:
17781:
17775:
17772:
17766:
17762:
17749:
17735:
17732:
17726:
17722:
17716:
17711:
17706:
17701:
17697:
17693:
17688:
17683:
17677:
17674:
17668:
17664:
17658:
17646:
17635:
17629:
17626:
17621:
17617:
17613:
17608:
17603:
17598:
17595:
17582:
17550:
17549:
17535:
17532:
17526:
17522:
17516:
17510:
17507:
17501:
17497:
17484:
17473:
17467:
17464:
17459:
17455:
17451:
17447:
17443:
17440:
17427:
17395:
17394:
17381:
17376:
17370:
17367:
17361:
17357:
17351:
17339:
17328:
17322:
17319:
17312:
17307:
17302:
17289:
17272:
17271:
17258:
17253:
17245:
17242:
17236:
17232:
17219:
17205:
17202:
17196:
17192:
17185:
17174:
17163:
17157:
17154:
17148:
17143:
17139:
17126:
17098:
17097:
17083:
17080:
17074:
17070:
17064:
17054:
17043:
17037:
17034:
17028:
17024:
17021:
17008:
16978:
16977:
16965:
16954:
16943:
16937:
16934:
16928:
16924:
16911:
16897:
16896:
16885:
16874:
16871:
16852:
16849:
16843:
16839:
16821:
16818:
16817:
16816:
16813:
16812:
16798:
16794:
16790:
16786:
16782:
16776:
16772:
16768:
16754:
16740:
16736:
16732:
16728:
16724:
16717:
16713:
16708:
16704:
16700:
16696:
16683:
16672:
16665:
16660:
16655:
16651:
16647:
16643:
16639:
16635:
16622:
16608:
16607:
16596:
16586:
16569:
16557:
16545:
16542:
16532:
16515:
16510:
16496:
16477:
16450:
16446:
16435:
16418:
16415:
16403:
16389:
16385:
16374:
16357:
16347:
16336:
16322:
16317:
16307:
16290:
16278:
16274:
16263:
16246:
16242:
16229:
16191:
16190:
16171:
16165:
16146:
16141:
16121:
16110:
16098:
16092:
16073:
16068:
16048:
16042:
16023:
16012:
15998:
15984:
15963:
15950:
15937:
15917:
15916:
15899:
15895:
15889:
15885:
15861:
15853:
15849:
15843:
15839:
15826:
15809:
15805:
15799:
15795:
15768:
15764:
15758:
15754:
15746:
15741:
15729:
15718:
15711:
15707:
15701:
15695:
15690:
15685:
15679:
15672:
15668:
15663:
15659:
15655:
15651:
15647:
15644:
15631:
15601:
15600:
15588:
15584:
15574:
15561:
15556:
15551:
15541:
15530:
15523:
15519:
15512:
15507:
15503:
15499:
15494:
15488:
15481:
15477:
15471:
15465:
15460:
15455:
15449:
15442:
15438:
15433:
15429:
15425:
15421:
15417:
15414:
15401:
15398:
15397:
15385:
15381:
15371:
15360:
15352:
15347:
15342:
15338:
15334:
15328:
15323:
15317:
15312:
15305:
15301:
15287:
15263:
15262:
15249:
15244:
15239:
15235:
15224:
15213:
15205:
15200:
15195:
15191:
15187:
15181:
15176:
15170:
15165:
15158:
15154:
15140:
15126:
15125:
15113:
15108:
15104:
15094:
15082:
15076:
15071:
15066:
15056:
15045:
15038:
15034:
15028:
15023:
15017:
15012:
15007:
14994:
14970:
14969:
14957:
14952:
14946:
14941:
14936:
14932:
14927:
14916:
14904:
14898:
14893:
14888:
14884:
14879:
14873:
14868:
14856:
14845:
14838:
14834:
14828:
14823:
14817:
14812:
14807:
14794:
14780:
14779:
14767:
14761:
14756:
14744:
14732:
14726:
14721:
14709:
14698:
14691:
14687:
14681:
14676:
14670:
14665:
14660:
14647:
14621:
14620:
14608:
14597:
14584:
14579:
14567:
14556:
14549:
14545:
14539:
14533:
14528:
14523:
14517:
14510:
14506:
14500:
14494:
14489:
14484:
14460:
14453:
14449:
14444:
14440:
14436:
14432:
14428:
14425:
14419:
14412:
14408:
14403:
14399:
14395:
14391:
14387:
14384:
14371:
14357:
14356:
14354:Hessian matrix
14340:
14329:
14316:
14311:
14299:
14288:
14280:
14275:
14270:
14266:
14262:
14257:
14252:
14248:
14234:
14231:
14230:
14213:
14209:
14203:
14199:
14193:
14186:
14182:
14176:
14172:
14148:
14142:
14137:
14128:
14124:
14118:
14114:
14108:
14104:
14099:
14091:
14087:
14081:
14077:
14064:
14047:
14043:
14037:
14033:
14027:
14020:
14016:
14010:
14006:
13979:
13975:
13969:
13965:
13957:
13952:
13945:
13940:
13935:
13928:
13924:
13918:
13914:
13907:
13901:
13896:
13884:
13873:
13866:
13862:
13856:
13851:
13845:
13840:
13835:
13829:
13822:
13818:
13813:
13809:
13804:
13800:
13796:
13792:
13789:
13776:
13731:
13730:
13713:
13709:
13703:
13699:
13693:
13686:
13682:
13676:
13672:
13648:
13640:
13636:
13630:
13626:
13620:
13616:
13608:
13604:
13598:
13594:
13581:
13564:
13560:
13554:
13550:
13544:
13537:
13533:
13527:
13523:
13496:
13492:
13486:
13482:
13474:
13469:
13464:
13457:
13453:
13447:
13443:
13435:
13430:
13418:
13407:
13400:
13396:
13390:
13384:
13379:
13374:
13368:
13361:
13357:
13352:
13348:
13344:
13340:
13336:
13333:
13320:
13288:
13287:
13272:
13268:
13263:
13260:
13251:
13248:
13243:
13240:
13237:
13234:
13231:
13222:
13219:
13214:
13211:
13208:
13205:
13202:
13189:
13178:
13171:
13167:
13162:
13159:
13156:
13153:
13150:
13147:
13144:
13141:
13128:
13111:
13110:
13095:
13091:
13086:
13083:
13074:
13071:
13066:
13063:
13060:
13057:
13054:
13041:
13030:
13023:
13019:
13014:
13011:
13008:
13005:
13002:
12989:
12972:
12971:
12956:
12952:
12947:
12944:
12938:
12935:
12928:
12924:
12919:
12916:
12910:
12900:
12889:
12882:
12878:
12873:
12870:
12867:
12854:
12822:
12821:
12806:
12802:
12797:
12794:
12788:
12781:
12777:
12772:
12769:
12756:
12745:
12738:
12734:
12729:
12726:
12723:
12720:
12717:
12714:
12701:
12669:
12668:
12653:
12649:
12644:
12641:
12635:
12625:
12614:
12607:
12603:
12598:
12595:
12592:
12579:
12549:
12548:
12536:
12525:
12512:
12507:
12495:
12484:
12477:
12473:
12468:
12465:
12452:
12438:
12437:
12426:
12415:
12412:
12396:
12390:
12385:
12381:
12374:
12370:
12365:
12362:
12341:
12338:
12337:
12336:
12333:
12332:
12317:
12313:
12308:
12304:
12300:
12296:
12292:
12282:
12278:
12273:
12269:
12265:
12261:
12257:
12247:
12243:
12237:
12233:
12220:
12205:
12201:
12195:
12191:
12181:
12177:
12172:
12168:
12164:
12160:
12156:
12146:
12142:
12137:
12133:
12129:
12125:
12121:
12108:
12097:
12090:
12086:
12081:
12078:
12074:
12070:
12066:
12062:
12058:
12054:
12041:
12024:
12023:
12008:
12004:
11999:
11995:
11991:
11987:
11983:
11973:
11969:
11963:
11959:
11946:
11931:
11927:
11921:
11917:
11907:
11903:
11898:
11894:
11890:
11886:
11882:
11869:
11858:
11851:
11847:
11842:
11838:
11834:
11830:
11826:
11813:
11796:
11795:
11780:
11776:
11770:
11766:
11760:
11753:
11749:
11743:
11739:
11726:
11715:
11708:
11704:
11699:
11695:
11691:
11687:
11683:
11680:
11667:
11635:
11634:
11621:
11616:
11607:
11603:
11597:
11593:
11580:
11565:
11561:
11555:
11551:
11544:
11533:
11522:
11515:
11511:
11505:
11500:
11496:
11483:
11453:
11452:
11439:
11434:
11425:
11421:
11416:
11413:
11407:
11400:
11396:
11390:
11386:
11380:
11370:
11355:
11351:
11346:
11343:
11336:
11332:
11325:
11321:
11315:
11311:
11305:
11295:
11284:
11277:
11273:
11267:
11263:
11260:
11247:
11215:
11214:
11201:
11196:
11187:
11183:
11178:
11175:
11162:
11147:
11143:
11138:
11135:
11128:
11117:
11106:
11099:
11095:
11089:
11085:
11082:
11069:
11039:
11038:
11023:
11019:
11013:
11009:
11003:
10993:
10982:
10975:
10970:
10964:
10960:
10957:
10944:
10914:
10913:
10901:
10890:
10877:
10872:
10860:
10849:
10842:
10838:
10832:
10826:
10821:
10816:
10803:
10789:
10788:
10775:
10770:
10758:
10746:
10735:
10724:
10717:
10713:
10707:
10702:
10698:
10685:
10671:
10670:
10658:
10647:
10636:
10629:
10625:
10619:
10615:
10602:
10599:
10598:
10586:
10575:
10564:
10557:
10553:
10547:
10543:
10530:
10516:
10515:
10501:
10487:
10484:
10464:
10460:
10454:
10450:
10428:
10425:
10424:
10423:
10364:the matrices,
10362:
10301:
10209:
10206:
10205:
10204:
10189:
10184:
10173:
10170:
10166:
10162:
10157:
10154:
10148:
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6988:Column vector
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6760:Column vector
6758:
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6727:estimate of a
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4171:tangent matrix
4154:
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4129:tangent matrix
4123:
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4093:
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3271:
3267:
3264:
3261:
3258:
3255:
3251:
3247:
3243:
3236:
3231:
3178:electric field
3174:
3173:
3159:
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3134:
3128:
3123:
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3103:
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2759:
2757:
2751:
2747:
2721:
2718:
2706:tangent vector
2662:
2659:
2653:
2649:
2631:of the vector
2628:tangent vector
2607:
2606:
2595:
2590:
2581:
2578:
2571:
2567:
2563:
2557:
2556:
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2334:
2331:
2265:Main article:
2262:
2259:
2241:
2238:
2237:
2236:
2232:
2067:matrices with
2032:
2029:
2028:
2027:
2022:
2017:
2012:
1998:
1995:
1977:
1974:
1955:
1954:
1952:
1950:
1935:
1931:
1926:
1923:
1910:
1906:
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1888:
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1473:
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1467:
1443:
1418:
1415:
1412:
1409:
1406:
1397:direction for
1384:
1380:
1357:
1350:
1347:
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1320:
1315:
1308:
1305:
1293:
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1187:
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1135:
1131:
1126:
1122:
1118:
1113:
1109:
1105:
1102:
1074:
1071:
1057:, statistics,
1014:, and/or of a
982:
981:
979:
978:
971:
964:
956:
953:
952:
949:
948:
943:
938:
933:
931:List of topics
928:
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912:
907:
906:
903:
902:
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898:
893:
888:
883:
877:
872:
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854:
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833:
830:
829:
824:
823:
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821:
816:
811:
806:
801:
796:
791:
783:
782:
778:
777:
776:
775:
770:
765:
760:
752:
751:
745:
738:
737:
734:
733:
728:
727:
726:
725:
720:
715:
710:
705:
700:
692:
691:
687:
686:
685:
684:
679:
674:
669:
664:
659:
649:
642:
641:
638:
637:
632:
631:
630:
629:
624:
619:
614:
609:
603:
598:
593:
588:
583:
575:
574:
568:
567:
566:
565:
560:
555:
550:
545:
540:
525:
518:
517:
514:
513:
508:
507:
506:
505:
500:
495:
490:
488:Changing order
485:
480:
475:
457:
452:
447:
439:
438:
437:Integration by
434:
433:
432:
431:
426:
421:
416:
411:
401:
399:Antiderivative
393:
392:
388:
387:
386:
385:
380:
375:
365:
358:
357:
354:
353:
348:
347:
346:
345:
340:
335:
330:
325:
320:
315:
310:
305:
300:
292:
291:
285:
284:
283:
282:
277:
272:
267:
262:
257:
249:
248:
244:
243:
242:
241:
240:
239:
234:
229:
219:
206:
205:
199:
192:
191:
188:
187:
185:
184:
179:
174:
168:
166:
165:
160:
154:
153:
152:
144:
143:
131:
128:
125:
122:
119:
116:
113:
110:
107:
104:
101:
98:
94:
91:
88:
84:
81:
75:
70:
66:
56:
53:
52:
46:
45:
26:
9:
6:
4:
3:
2:
36868:
36857:
36854:
36852:
36849:
36847:
36846:Matrix theory
36844:
36843:
36841:
36826:
36823:
36821:
36818:
36816:
36813:
36811:
36808:
36806:
36803:
36801:
36798:
36794:
36791:
36789:
36786:
36784:
36781:
36779:
36776:
36774:
36771:
36770:
36768:
36764:
36761:
36760:
36758:
36757:
36755:
36751:
36741:
36738:
36736:
36733:
36732:
36729:
36721:
36718:
36716:
36713:
36711:
36708:
36707:
36706:
36703:
36699:
36696:
36695:
36694:
36691:
36689:
36686:
36684:
36681:
36679:
36676:
36674:
36671:
36670:
36668:
36666:
36662:
36659:
36655:
36649:
36648:
36644:
36642:
36641:
36637:
36635:
36632:
36630:
36627:
36625:
36622:
36620:
36617:
36615:
36612:
36610:
36609:Infinitesimal
36607:
36605:
36602:
36600:
36597:
36595:
36592:
36590:
36587:
36585:
36582:
36581:
36579:
36577:
36573:
36567:
36564:
36562:
36559:
36557:
36554:
36552:
36549:
36547:
36544:
36543:
36541:
36535:
36527:
36524:
36522:
36519:
36517:
36514:
36512:
36509:
36507:
36504:
36502:
36499:
36497:
36494:
36492:
36489:
36487:
36484:
36482:
36479:
36478:
36476:
36472:
36469:
36465:
36462:
36460:
36457:
36456:
36455:
36452:
36450:
36447:
36445:
36442:
36440:
36437:
36435:
36432:
36430:
36427:
36425:
36422:
36421:
36419:
36417:
36414:
36413:
36411:
36407:
36399:
36396:
36394:
36391:
36389:
36386:
36384:
36381:
36380:
36378:
36376:
36373:
36371:
36368:
36366:
36363:
36361:
36358:
36356:
36353:
36351:
36350:Line integral
36348:
36346:
36343:
36341:
36338:
36336:
36333:
36331:
36328:
36326:
36323:
36322:
36320:
36318:
36314:
36306:
36303:
36301:
36298:
36296:
36293:
36291:
36288:
36287:
36285:
36281:
36278:
36276:
36273:
36271:
36268:
36266:
36263:
36261:
36258:
36257:
36255:
36254:
36252:
36250:
36246:
36240:
36237:
36235:
36232:
36228:
36225:
36223:
36222:Washer method
36220:
36219:
36217:
36215:
36212:
36208:
36205:
36204:
36203:
36200:
36196:
36193:
36191:
36188:
36186:
36185:trigonometric
36183:
36182:
36181:
36178:
36176:
36173:
36169:
36166:
36165:
36164:
36161:
36159:
36156:
36154:
36151:
36149:
36146:
36144:
36141:
36139:
36136:
36135:
36133:
36131:
36127:
36119:
36116:
36114:
36111:
36109:
36106:
36105:
36104:
36101:
36097:
36094:
36092:
36089:
36088:
36086:
36082:
36079:
36077:
36074:
36072:
36069:
36067:
36064:
36063:
36062:
36059:
36055:
36054:Related rates
36052:
36050:
36047:
36045:
36042:
36040:
36037:
36036:
36034:
36030:
36027:
36023:
36020:
36019:
36018:
36015:
36013:
36010:
36008:
36005:
36003:
36000:
35998:
35995:
35993:
35990:
35989:
35988:
35985:
35981:
35978:
35976:
35973:
35972:
35971:
35968:
35966:
35963:
35961:
35958:
35956:
35953:
35951:
35948:
35946:
35943:
35941:
35938:
35937:
35935:
35933:
35929:
35923:
35920:
35918:
35915:
35913:
35910:
35906:
35903:
35902:
35901:
35898:
35896:
35893:
35892:
35890:
35888:
35884:
35878:
35875:
35873:
35870:
35868:
35865:
35863:
35860:
35858:
35855:
35853:
35850:
35848:
35845:
35843:
35840:
35838:
35835:
35833:
35830:
35828:
35825:
35823:
35820:
35818:
35815:
35814:
35812:
35810:
35806:
35802:
35795:
35790:
35788:
35783:
35781:
35776:
35775:
35772:
35765:
35762:
35759:
35756:
35753:
35750:
35747:
35743:
35740:
35737:
35733:
35729:
35726:
35723:
35720:
35716:
35713:
35710:
35707:
35704:
35700:
35697:
35696:
35687:
35683:
35679:
35676:
35673:
35669:
35666:
35663:
35660:
35659:
35642:
35638:
35634:
35630:
35625:
35621:
35615:
35611:
35606:
35602:
35598:
35594:
35588:
35584:
35579:
35578:
35564:
35560:
35556:
35552:
35548:
35541:
35532:
35527:
35523:
35519:
35515:
35508:
35500:
35498:9781119541202
35494:
35490:
35483:
35475:
35469:
35465:
35458:
35450:
35448:9780387950532
35444:
35440:
35436:
35429:
35421:
35415:
35411:
35407:
35406:Fang, Kai-Tai
35401:
35394:
35389:
35381:
35377:
35373:
35367:
35363:
35359:
35355:
35347:
35340:
35334:
35315:
35308:
35290:
35274:
35259:
35242:
35236:
35211:
35196:
35189:
35188:
35180:
35178:
35176:
35174:
35172:
35170:
35168:
35166:
35164:
35162:
35160:
35158:
35156:
35154:
35152:
35150:
35148:
35140:
35136:
35132:
35118:
35111:
35104:
35088:
35081:
35079:
35077:
35075:
35073:
35068:
35043:
35037:
35034:
35029:
35026:
35023:
35020:
35014:
35008:
35005:
35000:
34997:
34994:
34991:
34985:
34979:
34976:
34971:
34968:
34965:
34962:
34956:
34950:
34947:
34939:
34936:
34933:
34930:
34927:
34924:
34921:
34915:
34909:
34903:
34900:
34895:
34892:
34889:
34886:
34883:
34860:
34854:
34851:
34846:
34843:
34840:
34837:
34831:
34825:
34822:
34817:
34814:
34806:
34803:
34798:
34792:
34789:
34784:
34781:
34775:
34769:
34766:
34762:
34757:
34751:
34748:
34740:
34737:
34731:
34722:
34719:
34715:
34710:
34704:
34701:
34696:
34693:
34690:
34687:
34684:
34672:
34668:The constant
34665:
34657:
34628:
34626:
34617:
34604:
34603:column vector
34576:
34574:
34572:
34567:
34557:
34554:
34552:
34549:
34547:
34544:
34542:
34539:
34538:
34534:
34528:
34523:
34516:
34514:
34510:
34506:
34501:
34499:
34495:
34491:
34458:
34452:
34449:
34439:
34429:
34415:
34412:
34403:
34395:
34388:
34387:
34367:
34356:
34346:
34336:
34317:
34308:
34300:
34293:
34292:
34272:
34266:
34263:
34253:
34243:
34229:
34226:
34217:
34209:
34202:
34201:
34181:
34170:
34165:
34162:
34152:
34130:
34118:
34115:
34112:
34109:
34106:
34099:
34098:
34071:
34060:
34055:
34052:
34042:
34023:
34008:
34005:
34002:
33995:
33994:
33979:
33976:
33970:
33967:
33962:
33959:
33949:
33935:
33932:
33928:
33925:
33922:
33919:
33912:
33911:
33907:
33904:
33903:
33897:
33896:
33895:
33893:
33890:
33873:
33861:
33858:
33854:
33850:
33845:
33833:
33803:
33791:
33787:
33781:
33777:
33773:
33746:
33731:
33727:
33720:
33715:
33711:
33707:
33693:
33670:
33664:
33657:
33630:
33608:
33604:
33600:
33595:
33592:
33557:
33554:
33534:
33521:
33516:
33510:
33487:
33484:
33474:
33471:
33456:
33453:
33437:
33432:
33429:
33419:
33401:
33383:
33380:
33376:
33339:
33335:
33331:
33326:
33322:
33311:
33307:
33303:
33298:
33294:
33283:
33279:
33272:
33269:
33261:
33257:
33250:
33238:
33234:
33230:
33225:
33221:
33210:
33206:
33198:
33195:
33188:
33181:
33163:
33155:
33143:
33140:
33136:
33128:
33114:
33110:
33095:
33091:
33087:
33080:
33078:
33063:
33059:
33050:
33045:
33026:
33014:
33011:
33007:
33003:
32998:
32986:
32970:
32952:
32940:
32936:
32930:
32926:
32922:
32910:
32909:
32894:
32891:
32885:
32882:
32868:
32865:
32846:
32838:
32835:
32821:
32818:
32800:
32796:
32788:
32774:
32770:
32765:
32761:
32758:
32754:
32750:
32743:
32741:
32740:
32725:
32722:
32708:
32705:
32702:
32695:
32683:
32670:
32666:
32660:
32655:
32651:
32643:
32629:
32625:
32614:
32610:
32606:
32599:
32597:
32596:
32579:
32576:
32573:
32570:
32567:
32549:
32541:
32529:
32526:
32523:
32518:
32515:
32512:
32508:
32500:
32486:
32482:
32477:
32467:
32463:
32456:
32450:
32449:
32417:
32407:
32393:
32389:
32372:
32368:
32361:
32358:
32354:
32353:
32336:
32333:
32322:
32313:
32309:
32303:
32300:
32290:
32283:
32269:
32265:
32260:
32257:
32247:
32243:
32236:
32234:
32233:
32203:
32193:
32179:
32175:
32160:
32156:
32149:
32147:
32146:
32123:
32117:
32109:
32101:
32090:
32080:
32066:
32055:
32044:
32037:
32034:
32030:
32029:
32006:
32000:
31992:
31984:
31973:
31963:
31949:
31938:
31927:
31920:
31917:
31913:
31912:
31889:
31878:
31862:
31852:
31838:
31819:
31812:
31810:
31809:
31789:
31786:
31778:
31771:
31757:
31746:
31735:
31728:
31726:
31725:
31705:
31701:
31694:
31680:
31669:
31663:
31656:
31653:
31648:
31645:
31644:
31629:
31622:
31608:
31594:
31587:
31584:
31579:
31576:
31575:
31571:
31568:
31565:
31564:
31558:
31557:
31537:
31528:
31523:
31520:
31509:
31505:
31502:
31495:
31481:
31460:
31457:
31451:
31444:
31443:
31420:
31406:
31403:
31397:
31394:
31391:
31387:
31378:
31373:
31370:
31359:
31355:
31352:
31330:
31316:
31292:
31285:
31284:
31261:
31255:
31252:
31245:
31231:
31214:
31211:
31205:
31198:
31197:
31193:
31190:
31189:
31183:
31182:
31181:
31180:is a scalar.
31178:
31148:
31137:
31133:
31123:
31119:
31116:
31095:
31089:
31080:
31070:
31066:
31062:
31058:
31055:
31042:
31039:
31033:
31029:
31028:
31012:
31000:
30993:
30978:
30974:
30971:
30950:
30944:
30925:
30911:
30908:
30895:
30890:
30883:
30876:
30869:
30862:
30858:
30851:
30843:
30835:
30828:
30821:
30817:
30811:
30805:
30801:
30800:
30784:
30777:
30751:
30724:
30715:
30709:
30705:
30702:
30695:
30680:
30673:
30630:
30620:
30616:
30613:
30606:
30592:
30586:
30567:
30554:
30550:
30546:
30542:
30538:
30537:
30521:
30516:
30511:
30506:
30499:
30478:
30475:
30464:
30459:
30455:
30452:
30449:
30445:
30438:
30417:
30414:
30403:
30399:
30394:
30390:
30386:
30382:
30373:
30369:
30353:
30340:
30337:
30326:
30322:
30319:
30315:
30310:
30302:
30280:
30272:
30268:
30242:
30227:
30223:
30219:
30215:
30211:
30210:
30194:
30187:
30166:
30163:
30152:
30148:
30145:
30124:
30118:
30095:
30092:
30079:
30075:
30071:
30067:
30063:
30062:
30046:
30039:
30018:
30015:
30004:
30000:
29997:
29961:
29955:
29922:
29918:
29914:
29910:
29906:
29905:
29901:
29895:
29889:Mixed layout,
29885:
29879:
29865:
29859:
29858:
29829:
29823:
29804:
29798:
29779:
29767:
29753:
29747:
29731:
29710:
29706:
29702:
29698:
29691:
29687:
29683:
29679:
29678:
29660:
29641:
29614:
29601:
29587:
29581:
29562:
29549:
29545:
29541:
29537:
29533:
29532:
29528:
29517:
29511:
29510:
29498:
29463:
29459:
29455:
29445:
29441:
29428:
29414:
29408:
29393:
29389:
29375:
29372:
29366:
29362:
29361:
29333:
29326:
29317:
29306:
29299:
29281:
29267:
29261:
29245:
29223:
29218:
29211:
29204:
29197:
29190:
29186:
29179:
29171:
29163:
29156:
29149:
29145:
29139:
29133:
29129:
29128:
29111:
29108:
29097:
29090:
29069:
29066:
29053:
29034:
29028:
29022:
29007:
28996:
28990:
28969:
28966:
28953:
28933:
28927:
28924:
28910:
28896:
28890:
28884:
28874:
28871:
28859:
28844:
28840:
28836:
28832:
28828:
28824:
28823:
28806:
28803:
28790:
28769:
28766:
28756:
28749:
28735:
28729:
28719:
28716:
28696:
28692:
28688:
28684:
28680:
28679:
28659:
28653:
28634:
28628:
28609:
28597:
28583:
28577:
28561:
28540:
28536:
28532:
28528:
28521:
28517:
28513:
28509:
28508:
28488:
28482:
28463:
28457:
28438:
28426:
28412:
28406:
28390:
28369:
28365:
28361:
28357:
28350:
28346:
28342:
28338:
28337:
28314:
28295:
28289:
28261:
28247:
28241:
28207:
28203:
28199:
28195:
28188:
28184:
28180:
28176:
28175:
28157:
28138:
28132:
28109:
28095:
28089:
28073:
28052:
28048:
28044:
28040:
28033:
28029:
28025:
28021:
28020:
27997:
27969:
27955:
27949:
27920:
27916:
27912:
27908:
27902:
27897:
27891:
27887:
27886:
27868:
27849:
27842:
27828:
27822:
27809:
27796:
27792:
27788:
27784:
27780:
27779:
27775:
27763:
27745:
27719:
27718:
27681:
27678:
27673:
27645:
27618:
27613:
27610:
27605:
27584:
27561:
27534:
27522:
27473:
27470:
27465:
27437:
27425:
27405:
27402:
27397:
27376:
27361:
27334:
27322:
27308:
27250:
27246:
27239:
27232:
27226:
27222:
27221:
27204:
27201:
27196:
27175:
27161:
27134:
27130:
27123:
27085:
27082:
27077:
27049:
27043:
27016:
27012:
27005:
26991:
26974:
26947:
26933:
26929:
26922:
26915:
26909:
26905:
26904:
26882:
26877:
26874:
26864:
26858:
26831:
26827:
26820:
26804:
26801:
26760:
26745:
26741:
26738:
26733:
26730:
26719:
26692:
26688:
26681:
26667:
26650:
26623:
26609:
26605:
26598:
26592:
26588:
26587:
26560:
26557:
26550:
26531:
26528:
26521:
26507:
26499:
26483:
26461:
26457:
26454:
26440:
26437:
26433:
26432:
26410:
26405:
26395:
26390:
26383:
26367:
26357:
26350:
26336:
26319:
26297:
26293:
26290:
26276:
26273:
26269:
26268:
26246:
26241:
26238:
26228:
26222:
26217:
26207:
26203:
26196:
26180:
26177:
26166:
26161:
26151:
26147:
26140:
26126:
26109:
26104:
26094:
26079:
26073:
26072:
26050:
26045:
26042:
26032:
26002:
25986:
25983:
25948:
25934:
25886:
25883:
25877:
25871:
25867:
25866:
25844:
25839:
25836:
25826:
25817:
25801:
25798:
25784:
25770:
25744:
25736:
25733:
25720:
25717:
25711:
25707:
25706:
25684:
25679:
25676:
25666:
25646:
25640:
25634:
25631:
25624:
25608:
25605:
25580:
25567:
25561:
25558:
25551:
25537:
25495:
25493:
25492:
25456:
25453:
25443:
25418:
25415:
25408:
25394:
25364:
25361:
25355:
25352:
25338:
25336:
25335:
25313:
25302:
25298:
25289:
25267:
25259:
25245:
25228:
25217:
25213:
25209:
25206:
25192:
25190:
25189:
25167:
25161:
25158:
25155:
25152:
25149:
25132:
25121:
25114:
25111:
25108:
25103:
25100:
25097:
25093:
25085:
25069:
25066:
25063:
25060:
25057:
25040:
25028:
25025:
25022:
25017:
25014:
25011:
25007:
24999:
24985:
24968:
24962:
24946:
24942:
24939:
24925:
24916:
24910:
24906:
24905:
24883:
24878:
24875:
24872:
24862:
24857:
24850:
24834:
24831:
24828:
24818:
24811:
24797:
24780:
24775:
24765:
24761:
24758:
24744:
24738:
24737:
24708:
24666:
24610:
24582:
24568:
24551:
24524:
24520:
24517:
24504:
24501:
24495:
24489:
24483:
24479:
24478:
24437:
24411:
24397:
24364:
24361:
24352:
24319:
24316:
24303:
24300:
24294:
24288:
24284:
24283:
24261:
24256:
24253:
24243:
24219:
24214:
24211:
24201:
24196:
24189:
24173:
24170:
24153:
24150:
24140:
24133:
24119:
24079:
24076:
24062:
24059:
24053:
24049:
24048:
24027:
24011:
24002:
23994:
23979:
23963:
23954:
23934:
23920:
23903:
23882:
23878:
23875:
23861:
23858:
23852:
23848:
23847:
23823:
23793:
23779:
23762:
23744:
23740:
23737:
23728:
23711:
23693:
23689:
23686:
23672:
23669:
23663:
23659:
23658:
23627:
23604:
23590:
23560:
23557:
23548:
23518:
23515:
23501:
23498:
23492:
23488:
23487:
23465:
23449:
23439:
23430:
23404:
23391:
23377:
23341:
23338:
23325:
23320:
23313:
23306:
23299:
23292:
23288:
23283:
23282:Taylor series
23277:
23269:
23261:
23254:
23249:
23243:
23239:
23235:
23234:
23195:
23192:
23183:
23162:
23138:
23132:
23129:
23116:
23112:
23108:
23104:
23097:
23091:
23087:
23086:
23047:
23044:
23035:
23008:
23005:
22978:
22954:
22943:
22940:
22927:
22923:
22919:
22915:
22908:
22904:
22900:
22896:
22895:
22857:
22830:
22827:
22814:
22812:
22811:
22762:
22728:
22707:
22669:
22638:
22629:
22615:
22545:
22542:
22536:
22530:
22524:
22520:
22519:
22484:
22467:
22451:
22442:
22434:
22413:
22389:
22372:
22356:
22347:
22342:
22333:
22319:
22295:
22259:
22233:
22230:
22224:
22218:
22212:
22208:
22207:
22171:
22136:
22122:
22067:
22064:
22058:
22052:
22048:
22047:
22011:
21976:
21962:
21914:
21911:
21905:
21899:
21895:
21894:
21891:
21888:
21869:
21861:
21858:
21854:
21829:
21824:
21808:
21799:
21796:
21792:
21765:
21738:
21729:
21723:
21719:
21716:
21709:
21694:
21685:
21682:
21678:
21634:
21624:
21620:
21617:
21610:
21596:
21588:
21585:
21581:
21561:
21543:
21539:
21535:
21530:
21502:
21490:
21479:
21473:
21461:
21450:
21444:
21417:
21395:
21389:
21383:
21370:
21366:
21362:
21358:
21354:
21353:
21325:
21313:
21302:
21296:
21269:
21250:
21244:
21231:
21227:
21223:
21219:
21215:
21214:
21186:
21177:
21174:
21158:
21149:
21128:
21112:
21109:
21096:
21092:
21088:
21084:
21077:
21073:
21069:
21065:
21064:
21036:
21027:
21011:
20984:
20965:
20962:
20959:
20943:
20939:
20935:
20931:
20924:
20920:
20916:
20912:
20911:
20883:
20874:
20853:
20837:
20834:
20821:
20817:
20813:
20809:
20803:
20797:
20793:
20792:
20768:
20738:
20724:
20708:
20695:
20692:
20686:
20682:
20681:
20677:
20669:
20657:
20629:
20613:
20612:
20611:
20609:
20589:
20551:
20534:
20517:
20490:
20486:
20483:
20470:
20455:
20431:
20386:
20369:
20342:
20338:
20335:
20322:
20321:
20320:
20317:
20299:
20290:
20280:
20241:
20223:
20217:
20213:
20210:
20207:
20205:
20196:
20187:
20183:
20156:
20110:
20105:
20101:
20098:
20095:
20093:
20084:
20072:
20042:
20038:
20035:
20032:
20028:
20016:
19967:
19963:
19960:
19957:
19955:
19946:
19919:
19904:
19900:
19897:
19894:
19890:
19836:
19829:
19825:
19822:
19819:
19817:
19808:
19781:
19766:
19762:
19759:
19756:
19752:
19742:
19723:
19702:
19697:
19693:
19690:
19687:
19685:
19676:
19649:
19645:
19630:
19627:
19624:
19620:
19601:
19580:
19576:
19573:
19570:
19568:
19559:
19532:
19517:
19513:
19510:
19507:
19503:
19498:
19483:
19479:
19461:
19457:
19454:
19451:
19449:
19440:
19404:
19400:
19396:
19393:
19390:
19386:
19381:
19363:
19359:
19344:
19340:
19337:
19334:
19332:
19323:
19289:
19285:
19282:
19278:
19256:
19241:
19237:
19234:
19231:
19229:
19220:
19215:
19188:
19184:
19180:
19176:
19173:
19170:
19166:
19139:
19135:
19132:
19129:
19126:
19124:
19089:
19086:
19083:
19060:
19014:
19011:
18958:
18955:
18952:
18929:
18926:
18923:
18900:
18897:
18894:
18892:
18867:
18864:
18856:
18841:
18837:
18834:
18831:
18829:
18813:
18810:
18799:
18798:
18797:
18795:
18790:
18786:
18776:
18670:
18636:
18630:
18611:
18607:
18601:
18582:
18578:
18559:
18542:
18523:
18515:
18507:
18501:
18478:
18464:
18458:
18442:
18421:
18417:
18413:
18409:
18402:
18398:
18394:
18390:
18389:
18384:
18296:
18273:
18256:
18163:
18149:
18143:
18094:
18090:
18086:
18081:
18076:
18023:
18000:
17983:
17925:
17911:
17905:
17867:
17863:
17859:
17854:
17832:
17826:
17807:
17802:
17787:
17779:
17773:
17750:
17733:
17714:
17699:
17691:
17681:
17675:
17656:
17647:
17633:
17627:
17611:
17583:
17579:
17575:
17571:
17564:
17560:
17556:
17552:
17551:
17533:
17514:
17508:
17471:
17465:
17449:
17428:
17424:
17420:
17416:
17409:
17405:
17401:
17397:
17396:
17374:
17368:
17349:
17326:
17320:
17290:
17286:
17282:
17278:
17274:
17273:
17243:
17220:
17203:
17175:
17161:
17155:
17127:
17123:
17119:
17115:
17109:
17104:
17100:
17099:
17081:
17062:
17041:
17035:
17022:
17009:
17005:
17001:
16997:
16990:
16984:
16980:
16979:
16941:
16935:
16912:
16909:
16903:
16899:
16898:
16891:
16880:
16868:
16850:
16824:
16823:
16788:
16770:
16755:
16730:
16702:
16684:
16670:
16645:
16623:
16620:
16614:
16610:
16609:
16584:
16543:
16530:
16478:
16433:
16413:
16372:
16348:
16334:
16305:
16261:
16230:
16227:
16221:
16215:
16209:
16203:
16197:
16193:
16192:
16169:
16144:
16119:
16111:
16096:
16071:
16046:
16024:
16010:
15938:
15935:
15929:
15923:
15919:
15918:
15915:
15827:
15825:
15730:
15716:
15677:
15653:
15632:
15628:
15624:
15620:
15613:
15607:
15603:
15602:
15582:
15575:
15549:
15542:
15528:
15510:
15486:
15447:
15423:
15402:
15400:
15399:
15379:
15358:
15303:
15288:
15286:
15281:
15275:
15269:
15265:
15264:
15237:
15211:
15156:
15141:
15138:
15132:
15128:
15127:
15102:
15095:
15064:
15057:
15043:
14995:
14993:
14988:
14982:
14976:
14972:
14971:
14950:
14934:
14925:
14917:
14902:
14886:
14877:
14857:
14843:
14795:
14792:
14786:
14782:
14781:
14745:
14710:
14696:
14648:
14645:
14639:
14633:
14627:
14623:
14622:
14598:
14568:
14554:
14515:
14458:
14434:
14417:
14393:
14372:
14369:
14363:
14359:
14358:
14355:
14330:
14300:
14286:
14255:
14250:
14235:
14233:
14232:
14229:
14191:
14106:
14065:
14063:
14025:
13933:
13885:
13871:
13827:
13798:
13777:
13774:
13768:
13760:
13756:
13752:
13745:
13741:
13737:
13733:
13732:
13729:
13691:
13618:
13582:
13580:
13542:
13462:
13419:
13405:
13366:
13342:
13321:
13317:
13313:
13309:
13302:
13298:
13294:
13290:
13289:
13261:
13249:
13238:
13232:
13220:
13209:
13203:
13176:
13154:
13148:
13142:
13129:
13125:
13121:
13117:
13113:
13112:
13084:
13072:
13061:
13055:
13028:
13009:
13003:
12990:
12986:
12982:
12978:
12974:
12973:
12945:
12936:
12933:
12917:
12908:
12887:
12871:
12868:
12855:
12851:
12847:
12843:
12836:
12832:
12828:
12824:
12823:
12795:
12786:
12770:
12743:
12724:
12721:
12718:
12702:
12698:
12694:
12690:
12683:
12679:
12675:
12671:
12670:
12642:
12633:
12612:
12596:
12593:
12580:
12576:
12572:
12568:
12561:
12555:
12551:
12550:
12526:
12496:
12482:
12466:
12453:
12450:
12444:
12440:
12439:
12434:
12423:
12409:
12394:
12379:
12363:
12347:
12346:
12345:
12221:
12109:
12095:
12042:
12038:
12034:
12030:
12026:
12025:
11947:
11870:
11856:
11814:
11810:
11806:
11802:
11798:
11797:
11758:
11713:
11689:
11668:
11664:
11660:
11656:
11649:
11645:
11641:
11637:
11636:
11581:
11534:
11520:
11484:
11480:
11476:
11472:
11465:
11459:
11455:
11454:
11414:
11405:
11378:
11371:
11344:
11330:
11303:
11296:
11282:
11261:
11248:
11244:
11240:
11236:
11229:
11225:
11221:
11217:
11216:
11176:
11163:
11136:
11118:
11104:
11083:
11070:
11067:
11061:
11053:
11049:
11045:
11041:
11040:
11001:
10980:
10958:
10945:
10941:
10937:
10933:
10926:
10920:
10916:
10915:
10891:
10861:
10847:
10804:
10801:
10795:
10791:
10790:
10759:
10736:
10722:
10686:
10683:
10677:
10673:
10672:
10634:
10603:
10601:
10600:
10562:
10531:
10528:
10522:
10518:
10517:
10513:
10507:
10499:
10493:
10481:
10435:
10434:
10433:
10420:
10414:
10404:
10398:
10392:
10386:
10380:
10374:
10368:
10363:
10359:
10353:
10343:
10337:
10331:
10325:
10319:
10313:
10307:
10303:the vectors,
10302:
10298:
10292:
10282:
10276:
10270:
10264:
10258:
10252:
10246:
10242:the scalars,
10241:
10240:
10239:
10236:
10234:
10230:
10226:
10222:
10218:
10213:
10187:
10182:
10171:
10168:
10164:
10155:
10144:
10134:
10131:
10127:
10118:
10102:
10099:
10095:
10086:
10073:
10068:
10063:
10058:
10046:
10043:
10039:
10030:
10019:
10009:
10005:
9996:
9980:
9976:
9967:
9949:
9946:
9942:
9933:
9922:
9912:
9908:
9899:
9883:
9879:
9870:
9858:
9853:
9851:
9833:
9820:
9815:
9804:
9800:
9789:
9785:
9773:
9763:
9759:
9748:
9744:
9727:
9723:
9712:
9708:
9694:
9689:
9684:
9679:
9667:
9663:
9652:
9648:
9636:
9626:
9622:
9611:
9607:
9590:
9586:
9575:
9571:
9552:
9548:
9537:
9533:
9521:
9511:
9507:
9496:
9492:
9475:
9471:
9460:
9456:
9443:
9438:
9436:
9403:
9398:
9389:
9379:
9375:
9363:
9355:
9345:
9341:
9326:
9316:
9312:
9299:
9294:
9292:
9284:
9261:
9256:
9245:
9241:
9232:
9219:
9207:
9203:
9194:
9176:
9172:
9163:
9151:
9146:
9144:
9126:
9109:
9108:
9107:
9081:
9076:
9068:
9065:
9061:
9057:
9052:
9045:
9042:
9038:
9034:
9027:
9024:
9020:
9016:
9009:
9004:
8999:
8994:
8985:
8982:
8978:
8974:
8969:
8962:
8958:
8954:
8947:
8943:
8939:
8930:
8927:
8923:
8919:
8914:
8907:
8903:
8899:
8892:
8888:
8884:
8878:
8873:
8871:
8861:
8854:
8849:
8840:
8830:
8827:
8823:
8811:
8803:
8793:
8790:
8786:
8771:
8761:
8758:
8754:
8740:
8735:
8730:
8725:
8715:
8705:
8702:
8698:
8686:
8678:
8668:
8664:
8649:
8639:
8635:
8618:
8608:
8605:
8601:
8589:
8581:
8571:
8567:
8552:
8542:
8538:
8525:
8520:
8518:
8510:
8483:
8482:
8481:
8460:
8455:
8444:
8441:
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8319:
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8312:
8303:
8292:
8282:
8278:
8269:
8253:
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8240:
8222:
8219:
8215:
8206:
8195:
8185:
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8172:
8156:
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8143:
8131:
8126:
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8106:
8093:
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8062:
8058:
8046:
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8032:
8021:
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7996:
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7419:
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7399:
7382:
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7380:
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7357:
7353:
7350:
7349:
7256:
7252:
7248:
7221:
7205:
7201:
7197:
7190:
7184:
7179:
7175:
7171:
7169:
7168:column vector
7162:
7159:
7158:
7109:
7105:
7101:
7057:
7050:
7023:
7007:
7003:
6999:
6992:
6986:
6983:
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6978:
6974:
6971:
6970:
6965:
6961:
6957:
6940:
6915:
6914:column vector
6908:
6891:
6848:
6840:
6824:
6821:
6815:
6811:
6808:
6805:
6802:
6799:
6796:
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6790:
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6770:
6764:
6756:
6748:
6742:
6741:
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6734:
6730:
6726:
6720:
6716:
6713:
6694:
6645:
6636:
6609:
6603:
6584:
6568:
6538:
6531:
6517:
6503:
6499:
6495:according to
6469:
6454:
6450:according to
6434:
6410:
6407:
6403:
6399:according to
6373:
6358:
6354:according to
6338:
6314:
6311:
6310:
6309:
6306:
6289:
6283:
6258:
6252:
6246:
6240:
6223:
6217:
6168:
6153:
6140:
6091:
6085:
6045:
6031:
6017:
5999:
5950:
5937:
5921:
5889:
5871:
5822:
5809:
5793:
5761:
5760:
5759:
5745:
5739:
5690:
5677:
5668:
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5614:
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5564:
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5532:
5518:
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5512:
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5322:
5316:
5310:
5304:
5300:
5294:
5290:
5285:
5280:
5274:
5268:
5221:
5212:
5203:
5201:
5200:Kalman filter
5197:
5193:
5174:
5170:
5148:
5138:
5134:
5131:
5128:
5125:
5106:
5105:
5104:
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4903:
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4639:
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4492:
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4260:
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4214:
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4150:
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4134:
4130:
4119:
4106:
4098:
4071:
4052:
4043:
4037:
4031:
4025:
4023:
4022:
4017:
4016:
4010:
4004:
3999:
3980:
3975:
3964:
3960:
3949:
3945:
3933:
3923:
3919:
3908:
3904:
3887:
3883:
3872:
3868:
3854:
3849:
3844:
3839:
3827:
3823:
3812:
3808:
3796:
3786:
3782:
3771:
3767:
3750:
3746:
3735:
3731:
3712:
3708:
3697:
3693:
3681:
3671:
3667:
3656:
3652:
3635:
3631:
3620:
3616:
3603:
3598:
3567:
3566:
3565:
3563:
3537:
3529:
3525:
3519:
3512:
3508:
3500:
3496:
3489:
3483:
3448:
3440:
3436:
3430:
3423:
3419:
3411:
3407:
3400:
3394:
3381:
3376:
3367:
3365:
3349:
3328:
3319:
3316:
3276:
3262:
3256:
3241:
3221:
3220:
3219:
3216:
3210:
3203:
3199:
3194:
3189:
3187:
3183:
3179:
3151:
3135:
3126:
3121:
3116:
3105:
3101:
3092:
3079:
3067:
3063:
3054:
3042:
3037:
3034:
3024:
3023:
3022:
3019:
3013:
3009:
3005:
3000:
2996:
2977:
2972:
2960:
2956:
2947:
2935:
2924:
2920:
2911:
2893:
2889:
2880:
2867:
2862:
2846:
2833:
2832:
2831:
2829:
2803:
2795:
2791:
2785:
2778:
2774:
2766:
2762:
2755:
2749:
2735:
2731:
2727:
2717:
2715:
2711:
2707:
2703:
2699:
2695:
2691:
2688:
2684:
2680:
2660:
2635:
2630:
2629:
2623:
2617:
2612:
2593:
2588:
2579:
2576:
2569:
2565:
2561:
2551:
2541:
2538:
2531:
2527:
2523:
2510:
2507:
2500:
2496:
2492:
2483:
2478:
2472:
2469:
2459:
2449:
2448:
2447:
2445:
2440:
2436:
2410:
2402:
2398:
2392:
2385:
2381:
2373:
2369:
2362:
2356:
2344:
2340:
2330:
2328:
2324:
2320:
2316:
2313:
2306:
2300:
2296:
2286:
2282:
2277:
2272:
2268:
2258:
2256:
2251:
2247:
2233:
2229:
2228:
2227:
2225:
2221:
2217:
2213:
2209:
2206:
2202:
2198:
2191:
2182:
2176:
2170:
2164:
2159:
2153:
2147:
2141:
2134:
2128:
2122:
2116:
2111:
2110:column vector
2108:, that is, a
2105:
2101:
2095:
2089:
2083:
2077:
2071:
2065:
2061:
2057:
2051:
2047:
2043:
2038:
2026:
2023:
2021:
2018:
2016:
2015:Wiener filter
2013:
2011:
2010:Kalman filter
2008:
2007:
2006:
2004:
1994:
1992:
1987:
1983:
1973:
1971:
1966:
1961:
1924:
1911:
1907:
1862:
1835:
1822:
1818:
1800:
1777:
1760:
1737:
1720:
1712:
1699:
1695:
1681:
1675:
1671:
1668:
1664:
1651:
1632:
1619:
1607:
1603:
1594:
1576:
1572:
1563:
1545:
1541:
1532:
1519:
1513:
1501:
1485:
1476:
1471:
1468:
1458:
1457:
1456:
1432:
1416:
1413:
1410:
1407:
1404:
1382:
1378:
1355:
1345:
1318:
1313:
1303:
1291:
1287:
1278:
1269:
1264:
1254:
1242:
1238:
1229:
1220:
1215:
1205:
1193:
1189:
1180:
1171:
1168:
1158:
1157:
1156:
1137:
1133:
1129:
1124:
1120:
1116:
1111:
1107:
1100:
1092:
1088:
1083:
1081:
1070:
1068:
1064:
1060:
1056:
1052:
1048:
1043:
1041:
1037:
1033:
1029:
1025:
1021:
1017:
1013:
1009:
1005:
1001:
997:
993:
989:
977:
972:
970:
965:
963:
958:
957:
955:
954:
947:
944:
942:
939:
937:
934:
932:
929:
927:
924:
922:
919:
917:
914:
913:
905:
904:
897:
894:
892:
889:
887:
884:
882:
879:
878:
870:
869:
858:
855:
853:
850:
848:
845:
844:
843:
842:
832:
831:
820:
817:
815:
812:
810:
807:
805:
802:
800:
799:Line integral
797:
795:
792:
790:
787:
786:
785:
784:
780:
779:
774:
771:
769:
766:
764:
761:
759:
756:
755:
754:
753:
749:
748:
742:
741:Multivariable
736:
735:
724:
721:
719:
716:
714:
711:
709:
706:
704:
701:
699:
696:
695:
694:
693:
689:
688:
683:
680:
678:
675:
673:
670:
668:
665:
663:
660:
658:
655:
654:
653:
652:
646:
640:
639:
628:
625:
623:
620:
618:
615:
613:
610:
608:
604:
602:
599:
597:
594:
592:
589:
587:
584:
582:
579:
578:
577:
576:
573:
570:
569:
564:
561:
559:
556:
554:
551:
549:
546:
544:
541:
538:
534:
531:
530:
529:
528:
522:
516:
515:
504:
501:
499:
496:
494:
491:
489:
486:
484:
481:
479:
476:
473:
469:
465:
464:trigonometric
461:
458:
456:
453:
451:
448:
446:
443:
442:
441:
440:
436:
435:
430:
427:
425:
422:
420:
417:
415:
412:
409:
405:
402:
400:
397:
396:
395:
394:
390:
389:
384:
381:
379:
376:
374:
371:
370:
369:
368:
362:
356:
355:
344:
341:
339:
336:
334:
331:
329:
326:
324:
321:
319:
316:
314:
311:
309:
306:
304:
301:
299:
296:
295:
294:
293:
290:
287:
286:
281:
278:
276:
275:Related rates
273:
271:
268:
266:
263:
261:
258:
256:
253:
252:
251:
250:
246:
245:
238:
235:
233:
232:of a function
230:
228:
227:infinitesimal
225:
224:
223:
220:
217:
213:
210:
209:
208:
207:
203:
202:
196:
190:
189:
183:
180:
178:
175:
173:
170:
169:
164:
161:
159:
156:
155:
151:
148:
147:
146:
145:
126:
120:
117:
111:
105:
102:
99:
96:
89:
82:
79:
73:
68:
64:
55:
54:
51:
48:
47:
43:
42:
37:
33:
19:
36720:Secant cubed
36645:
36638:
36619:Isaac Newton
36589:Brook Taylor
36354:
36256:Derivatives
36227:Shell method
35955:Differential
35746:Econometrics
35632:
35628:
35609:
35582:
35546:
35540:
35521:
35517:
35507:
35488:
35482:
35463:
35457:
35438:
35435:Fang, Kaitai
35428:
35409:
35400:
35388:
35353:
35346:
35333:
35321:. Retrieved
35307:
35257:
35209:
35199:. Retrieved
35195:the original
35186:
35121:. Retrieved
35116:
35103:
35091:. Retrieved
34670:
34664:
34655:
34615:
34601:refers to a
34502:
34487:
34484:Applications
33894:
33891:
33520:eigenvectors
33514:
33508:
33367:
33043:
32972:
31651:
31646:
31582:
31577:
31176:
31173:
31037:
31031:
30888:
30881:
30874:
30867:
30860:
30856:
30849:
30841:
30833:
30826:
30819:
30815:
30809:
30803:
30548:
30544:
30540:
30221:
30217:
30213:
30073:
30069:
30065:
29916:
29912:
29908:
29899:
29893:
29883:
29877:
29704:
29700:
29696:
29689:
29685:
29681:
29543:
29539:
29535:
29370:
29364:
29216:
29209:
29202:
29195:
29188:
29184:
29177:
29169:
29161:
29154:
29147:
29143:
29137:
29131:
28838:
28834:
28830:
28826:
28690:
28686:
28682:
28534:
28530:
28526:
28519:
28515:
28511:
28363:
28359:
28355:
28348:
28344:
28340:
28201:
28197:
28193:
28186:
28182:
28178:
28046:
28042:
28038:
28031:
28027:
28023:
27914:
27910:
27906:
27900:
27895:
27889:
27790:
27786:
27782:
27773:
27244:
27237:
27230:
27224:
26931:is symmetric
26927:
26920:
26913:
26907:
26603:
26596:
26590:
25881:
25875:
25869:
25715:
25709:
24914:
24908:
24499:
24493:
24487:
24481:
24298:
24292:
24286:
24057:
24051:
23856:
23850:
23667:
23661:
23496:
23490:
23318:
23311:
23304:
23297:
23290:
23286:
23275:
23267:
23259:
23252:
23241:
23237:
23110:
23106:
23102:
23095:
23089:
22921:
22917:
22913:
22906:
22902:
22898:
22540:
22534:
22528:
22522:
22228:
22222:
22216:
22210:
22062:
22056:
22050:
21909:
21903:
21897:
21886:
21883:
21827:
21541:
21537:
21533:
21364:
21360:
21356:
21225:
21221:
21217:
21090:
21086:
21082:
21075:
21071:
21067:
20937:
20933:
20929:
20922:
20918:
20914:
20815:
20811:
20807:
20801:
20795:
20690:
20684:
20675:
20667:
20605:
20318:
18997:
18785:product rule
18782:
18668:
18667:
18415:
18411:
18407:
18400:
18396:
18392:
18092:
18088:
18084:
17865:
17861:
17857:
17577:
17573:
17569:
17562:
17558:
17554:
17422:
17418:
17414:
17407:
17403:
17399:
17284:
17280:
17276:
17121:
17117:
17113:
17107:
17102:
17003:
16999:
16995:
16988:
16982:
16907:
16901:
16889:
16878:
16618:
16612:
16225:
16219:
16213:
16207:
16201:
16195:
15933:
15927:
15921:
15873:
15783:
15626:
15622:
15618:
15611:
15605:
15279:
15273:
15267:
15136:
15130:
14986:
14980:
14974:
14790:
14784:
14643:
14637:
14631:
14625:
14367:
14361:
14160:
13994:
13772:
13766:
13758:
13754:
13750:
13743:
13739:
13735:
13660:
13511:
13315:
13311:
13307:
13300:
13296:
13292:
13123:
13119:
13115:
12984:
12980:
12976:
12849:
12845:
12841:
12834:
12830:
12826:
12696:
12692:
12688:
12681:
12677:
12673:
12574:
12570:
12566:
12559:
12553:
12448:
12442:
12432:
12421:
12343:
12036:
12032:
12028:
11808:
11804:
11800:
11662:
11658:
11654:
11647:
11643:
11639:
11478:
11474:
11470:
11463:
11457:
11242:
11238:
11234:
11227:
11223:
11219:
11065:
11059:
11051:
11047:
11043:
10939:
10935:
10931:
10924:
10918:
10799:
10793:
10681:
10675:
10526:
10520:
10511:
10505:
10497:
10491:
10430:
10418:
10412:
10402:
10396:
10390:
10384:
10378:
10372:
10366:
10357:
10351:
10341:
10335:
10329:
10323:
10317:
10311:
10305:
10296:
10290:
10280:
10274:
10268:
10262:
10256:
10250:
10244:
10237:
10228:
10221:product rule
10214:
10211:
9105:
8479:
7378:
7370:
7359:
7355:
7351:Denominator
7254:
7250:
7199:
7195:
7188:
7177:
7173:
7160:Denominator
7107:
7103:
7001:
6997:
6990:
6976:
6972:Denominator
6963:
6959:
6819:
6788:
6784:
6778:
6768:
6762:
6754:
6736:
6732:
6721:
6717:
6554:
6497:
6452:
6409:Mixed layout
6408:
6401:
6356:
6312:
6304:
6256:
6250:
6244:
6238:
6154:
6107:
5673:
5666:
5608:
5562:
5514:
5510:
5506:
5502:
5498:
5493:
5487:
5482:
5475:
5365:
5361:
5356:
5351:
5345:
5339:
5334:
5326:
5324:elements of
5320:
5314:
5312:elements of
5308:
5302:
5298:
5292:
5288:
5283:
5278:
5272:
5266:
5222:
5218:
5209:
5189:
5103:is given by
5099:
5093:
5091:of a matrix
5086:
5082:
5080:of a scalar
5077:
5071:
4982:
4971:
4582:
4576:
4572:
4566:
4563:
4170:
4165:
4163:by a scalar
4159:
4156:
4138:
4137:
4132:
4128:
4125:
4045:is given by
4041:
4035:
4029:
4026:
4019:
4013:
4008:
4002:
3995:
3377:
3373:
3296:
3214:
3208:
3201:
3197:
3190:
3175:
3017:
3011:
3007:
3003:
2992:
2737:by a vector
2733:
2723:
2714:acceleration
2693:
2692:
2686:
2682:
2678:
2633:
2626:
2621:
2615:
2608:
2438:
2336:
2318:
2317:
2311:
2304:
2298:
2284:
2280:
2273:
2270:
2243:
2240:Alternatives
2211:
2210:
2204:
2189:
2180:
2168:
2157:
2151:
2145:
2139:
2132:
2126:
2120:
2114:
2103:
2099:
2093:
2087:
2081:
2075:
2069:
2063:
2059:
2049:
2045:
2041:
2034:
2000:
1979:
1962:
1958:
1672:
1666:
1662:
1647:
1430:
1333:
1084:
1076:
1055:econometrics
1044:
1034:, while the
1006:of a single
991:
985:
757:
460:Substitution
222:Differential
195:Differential
36788:of surfaces
36539:and numbers
36501:Dirichlet's
36471:Telescoping
36424:Alternating
36012:L'Hôpital's
35809:Precalculus
35693:Information
35672:Mathematica
29870:Expression
29527:dot product
29522:Expression
27768:Expression
21830:layout for
20662:Expression
20319:Therefore,
16873:Expression
12414:Expression
10486:Expression
5270:is of size
4978:determinant
2197:determinant
1032:engineering
988:mathematics
916:Precalculus
909:Miscellanea
874:Specialized
781:Definitions
548:Alternating
391:Definitions
204:Definitions
36840:Categories
36584:Adequality
36270:Divergence
36143:Arc length
35940:Derivative
35524:: 104849.
35419:3540176519
35323:5 February
35201:5 February
35123:5 February
35093:5 February
35063:References
34496:and other
31569:Expression
31191:Expression
29867:Condition
29519:Condition
29495:See also:
27765:Condition
23248:polynomial
20659:Condition
20610:section.)
18789:chain rule
16870:Condition
12411:Condition
10483:Condition
10217:chain rule
10208:Identities
7206:Numerator
7056:row vector
7008:Numerator
6980:row vector
6825:Numerator
5296:matrix or
5196:derivation
4127:the names
3364:derivative
2726:derivative
2700:vector in
2339:derivative
1028:statistics
896:Variations
891:Stochastic
881:Fractional
750:Formalisms
713:Divergence
682:Identities
662:Divergence
212:Derivative
163:Continuity
36783:of curves
36778:Curvature
36665:Integrals
36459:Maclaurin
36439:Geometric
36330:Geometric
36280:Laplacian
35992:linearity
35832:Factorial
35668:NCAlgebra
35601:569411497
35563:263661094
35280:∂
35272:∂
35234:∂
35224:∂
35135:transpose
35129:Uses the
35027:
34998:
34969:
34937:
34925:
34890:
34873:or, also
34844:
34691:
34119:
34082:⊤
34019:⊤
33855:δ
33788:λ
33778:∑
33728:λ
33712:∑
33605:λ
33584:Λ
33555:−
33544:Λ
33472:−
33377:δ
33336:λ
33332:≠
33323:λ
33308:λ
33304:−
33295:λ
33280:λ
33270:−
33258:λ
33235:λ
33222:λ
33207:λ
33137:∑
33060:λ
33008:δ
32937:λ
32927:∑
32883:−
32836:−
32806:∞
32797:∫
32762:
32706:−
32652:∫
32577:−
32571:−
32527:−
32509:∑
32334:−
32301:−
32291:−
32258:−
32217:⊤
32171:⊤
32118:∘
32102:∘
32056:∘
32001:⊗
31985:⊗
31939:⊗
31566:Condition
31521:−
31506:
31461:
31407:
31398:
31371:−
31356:
31256:
31215:
31120:
31087:∂
31059:
31053:∂
30975:
30942:∂
30912:
30906:∂
30854:, etc.);
30775:∂
30765:∂
30757:⊤
30741:∂
30722:∂
30706:
30671:∂
30661:∂
30647:∂
30628:∂
30617:
30584:∂
30565:∂
30497:∂
30487:∂
30476:−
30456:
30450:−
30436:∂
30426:∂
30415:−
30400:
30366:∂
30350:∂
30338:−
30323:
30265:∂
30239:∂
30185:∂
30175:∂
30164:−
30149:
30116:∂
30096:
30090:∂
30037:∂
30027:∂
30016:−
30001:
29953:∂
29933:∂
29830:⋅
29821:∂
29811:∂
29796:∂
29786:∂
29780:⋅
29745:∂
29732:⋅
29721:∂
29658:∂
29648:∂
29642:⋅
29631:∂
29612:∂
29579:∂
29560:∂
29406:∂
29386:∂
29259:∂
29234:∂
29182:, etc.);
29109:−
29088:∂
29078:∂
29067:−
29051:∂
29041:∂
29026:∂
29020:∂
29004:∂
28997:−
28988:∂
28978:∂
28967:−
28951:∂
28941:∂
28925:−
28888:∂
28882:∂
28872:−
28856:∂
28804:−
28788:∂
28778:∂
28767:−
28757:−
28727:∂
28717:−
28707:∂
28660:∘
28651:∂
28641:∂
28626:∂
28616:∂
28610:∘
28575:∂
28562:∘
28551:∂
28489:⊗
28480:∂
28470:∂
28455:∂
28445:∂
28439:⊗
28404:∂
28391:⊗
28380:∂
28312:∂
28302:∂
28287:∂
28277:∂
28239:∂
28218:∂
28155:∂
28145:∂
28130:∂
28120:∂
28087:∂
28063:∂
27995:∂
27985:∂
27947:∂
27931:∂
27866:∂
27856:∂
27820:∂
27807:∂
27743:∂
27733:∂
27679:−
27665:⊤
27655:⊤
27635:⊤
27611:−
27594:⊤
27545:⊤
27495:⊤
27485:⊤
27471:−
27457:⊤
27447:⊤
27417:⊤
27403:−
27386:⊤
27345:⊤
27298:∂
27275:⊤
27261:∂
27202:−
27185:⊤
27145:⊤
27107:⊤
27097:⊤
27083:−
27069:⊤
27059:⊤
27027:⊤
26981:∂
26958:⊤
26944:∂
26888:⊤
26875:−
26842:⊤
26802:−
26755:⊤
26731:−
26703:⊤
26657:∂
26634:⊤
26620:∂
26571:⊤
26558:−
26529:−
26490:∂
26473:⊤
26458:
26452:∂
26416:⊤
26326:∂
26309:⊤
26294:
26288:∂
26252:⊤
26239:−
26178:−
26116:∂
26091:∂
26056:⊤
26043:−
25984:−
25924:∂
25898:∂
25850:⊤
25837:−
25799:−
25760:∂
25737:
25731:∂
25690:⊤
25677:−
25635:
25606:−
25576:⊤
25562:
25527:∂
25507:∂
25476:⊤
25457:
25419:
25384:∂
25365:
25356:
25350:∂
25319:⊤
25235:∂
25210:
25204:∂
25173:⊤
25159:−
25153:−
25112:−
25094:∑
25067:−
25061:−
25026:−
25008:∑
24975:∂
24943:
24937:∂
24889:⊤
24876:−
24832:−
24787:∂
24762:
24756:∂
24703:⊤
24690:⊤
24680:⊤
24650:⊤
24640:⊤
24630:⊤
24620:⊤
24599:⊤
24558:∂
24543:⊤
24521:
24515:∂
24461:⊤
24451:⊤
24387:∂
24365:
24359:∂
24342:∂
24320:
24314:∂
24267:⊤
24254:−
24237:⊤
24225:⊤
24212:−
24197:−
24171:−
24151:−
24141:−
24109:∂
24092:−
24080:
24074:∂
24022:⊤
23974:⊤
23949:⊤
23910:∂
23892:⊤
23879:
23873:∂
23808:⊤
23769:∂
23754:⊤
23741:
23735:∂
23718:∂
23706:⊤
23690:
23684:∂
23642:⊤
23580:∂
23561:
23555:∂
23538:∂
23519:
23513:∂
23471:⊤
23367:∂
23342:
23336:∂
23212:∂
23196:
23190:∂
23152:∂
23133:
23127:∂
23064:∂
23048:
23042:∂
23025:∂
23009:
23003:∂
22968:∂
22944:
22938:∂
22847:∂
22831:
22825:∂
22795:⊤
22773:⊤
22758:⊤
22713:⊤
22702:⊤
22680:⊤
22665:⊤
22605:∂
22577:⊤
22556:∂
22503:⊤
22462:⊤
22419:⊤
22408:⊤
22367:⊤
22309:∂
22273:⊤
22244:∂
22191:⊤
22156:⊤
22112:∂
22100:⊤
22088:⊤
22078:∂
22031:⊤
21996:⊤
21952:∂
21935:⊤
21925:∂
21851:∂
21841:∂
21828:numerator
21789:∂
21779:∂
21771:⊤
21755:∂
21736:∂
21720:
21675:∂
21665:∂
21651:∂
21632:∂
21621:
21578:∂
21559:∂
21508:∂
21500:∂
21488:∂
21471:∂
21459:∂
21442:∂
21407:∂
21381:∂
21331:∂
21323:∂
21311:∂
21294:∂
21259:∂
21242:∂
21192:∂
21184:∂
21164:∂
21156:∂
21118:∂
21107:∂
21042:∂
21034:∂
21017:∂
21009:∂
20974:∂
20954:∂
20889:∂
20881:∂
20843:∂
20832:∂
20753:⊤
20714:∂
20706:∂
20635:∂
20627:∂
20584:⊤
20571:⊤
20561:⊤
20524:∂
20509:⊤
20487:
20481:∂
20444:⊤
20426:⊤
20416:⊤
20406:⊤
20396:⊤
20376:∂
20361:⊤
20339:
20333:∂
20286:⊤
20274:⊤
20261:⊤
20251:⊤
20214:
20169:⊤
20151:⊤
20141:⊤
20131:⊤
20121:⊤
20102:
20056:⊤
20039:
20008:⊤
19998:⊤
19988:⊤
19978:⊤
19964:
19940:⊤
19901:
19884:⊤
19874:⊤
19864:⊤
19854:⊤
19826:
19802:⊤
19763:
19748:⊤
19737:⊤
19694:
19670:⊤
19631:
19615:⊤
19577:
19553:⊤
19514:
19493:⊤
19458:
19434:⊤
19397:
19376:⊤
19341:
19317:⊤
19272:⊤
19238:
19210:⊤
19177:
19161:⊤
19136:
19111:⊤
19090:
19050:∂
19036:⊤
19015:
19009:∂
18959:
18930:
18901:
18868:
18851:⊤
18838:
18814:
18755:∂
18734:∂
18703:∂
18682:∂
18647:⊤
18637:×
18628:∂
18618:∂
18599:∂
18589:∂
18579:×
18574:⊤
18540:∂
18530:∂
18524:×
18508:×
18499:∂
18489:∂
18456:∂
18443:×
18432:∂
18362:∂
18341:∂
18327:∂
18306:∂
18294:∂
18284:∂
18254:∂
18244:∂
18230:∂
18209:∂
18195:∂
18174:∂
18141:∂
18109:∂
18054:∂
18033:∂
18021:∂
18011:∂
17981:∂
17971:∂
17957:∂
17936:∂
17903:∂
17882:∂
17838:⊤
17824:∂
17814:∂
17803:×
17798:⊤
17780:×
17771:∂
17761:∂
17731:∂
17721:∂
17715:×
17710:⊤
17692:×
17687:⊤
17673:∂
17663:∂
17625:∂
17612:×
17607:⊤
17594:∂
17531:∂
17521:∂
17506:∂
17496:∂
17463:∂
17439:∂
17380:⊤
17366:∂
17356:∂
17318:∂
17311:⊤
17301:∂
17257:⊤
17241:∂
17231:∂
17201:∂
17191:∂
17153:∂
17138:∂
17079:∂
17069:∂
17033:∂
17020:∂
16933:∂
16923:∂
16848:∂
16838:∂
16797:‖
16789:−
16781:‖
16771:−
16739:‖
16731:−
16723:‖
16716:⊤
16703:−
16659:∂
16654:‖
16646:−
16638:‖
16634:∂
16556:⊤
16509:⊤
16495:⊤
16449:⊤
16402:⊤
16388:⊤
16321:∂
16277:⊤
16241:∂
16164:⊤
16140:⊤
16091:⊤
16067:⊤
16041:⊤
15997:∂
15983:⊤
15962:⊤
15949:∂
15894:∂
15884:∂
15848:∂
15838:∂
15804:∂
15794:∂
15763:∂
15753:∂
15745:⊤
15706:∂
15694:⊤
15684:∂
15667:∂
15654:⋅
15643:∂
15560:⊤
15518:∂
15493:∂
15476:∂
15464:⊤
15454:∂
15437:∂
15424:⋅
15413:∂
15351:⊤
15341:∂
15333:∂
15316:⊤
15300:∂
15285:symmetric
15248:⊤
15204:⊤
15194:∂
15186:∂
15169:⊤
15153:∂
15075:⊤
15033:∂
15016:⊤
15006:∂
14992:symmetric
14945:⊤
14897:⊤
14872:⊤
14833:∂
14816:⊤
14806:∂
14760:⊤
14725:⊤
14686:∂
14669:⊤
14659:∂
14583:⊤
14544:∂
14532:⊤
14522:∂
14505:∂
14493:⊤
14483:∂
14448:∂
14435:⋅
14424:∂
14407:∂
14394:⋅
14383:∂
14315:⊤
14279:⊤
14269:∂
14261:∂
14247:∂
14208:∂
14198:∂
14181:∂
14171:∂
14141:⊤
14123:∂
14113:∂
14086:∂
14076:∂
14042:∂
14032:∂
14015:∂
14005:∂
13974:∂
13964:∂
13956:⊤
13944:⊤
13923:∂
13913:∂
13900:⊤
13861:∂
13844:⊤
13834:∂
13817:∂
13799:⋅
13788:∂
13708:∂
13698:∂
13681:∂
13671:∂
13635:∂
13625:∂
13603:∂
13593:∂
13559:∂
13549:∂
13532:∂
13522:∂
13491:∂
13481:∂
13473:⊤
13452:∂
13442:∂
13434:⊤
13395:∂
13383:⊤
13373:∂
13356:∂
13343:⋅
13332:∂
13267:∂
13259:∂
13247:∂
13230:∂
13218:∂
13201:∂
13166:∂
13140:∂
13090:∂
13082:∂
13070:∂
13053:∂
13018:∂
13001:∂
12951:∂
12943:∂
12923:∂
12915:∂
12877:∂
12866:∂
12801:∂
12793:∂
12776:∂
12768:∂
12733:∂
12713:∂
12648:∂
12640:∂
12602:∂
12591:∂
12511:⊤
12472:∂
12464:∂
12384:∇
12369:∂
12361:∂
12312:∂
12291:∂
12277:∂
12256:∂
12242:∂
12232:∂
12200:∂
12190:∂
12176:∂
12155:∂
12141:∂
12120:∂
12085:∂
12053:∂
12003:∂
11982:∂
11968:∂
11958:∂
11926:∂
11916:∂
11902:∂
11881:∂
11846:∂
11825:∂
11775:∂
11765:∂
11748:∂
11738:∂
11703:∂
11679:∂
11620:⊤
11602:∂
11592:∂
11560:∂
11550:∂
11510:∂
11495:∂
11438:⊤
11420:∂
11412:∂
11395:∂
11385:∂
11350:∂
11342:∂
11320:∂
11310:∂
11272:∂
11259:∂
11200:⊤
11182:∂
11174:∂
11142:∂
11134:∂
11094:∂
11081:∂
11018:∂
11008:∂
10969:∂
10956:∂
10876:⊤
10837:∂
10825:⊤
10815:∂
10774:⊤
10712:∂
10697:∂
10624:∂
10614:∂
10552:∂
10542:∂
10459:∂
10449:∂
10161:∂
10153:∂
10145:⋯
10124:∂
10116:∂
10092:∂
10084:∂
10074:⋮
10069:⋱
10064:⋮
10059:⋮
10036:∂
10028:∂
10020:⋯
10002:∂
9994:∂
9973:∂
9965:∂
9939:∂
9931:∂
9923:⋯
9905:∂
9897:∂
9876:∂
9868:∂
9839:∂
9831:∂
9797:∂
9782:∂
9774:⋯
9756:∂
9741:∂
9720:∂
9705:∂
9695:⋮
9690:⋱
9685:⋮
9680:⋮
9660:∂
9645:∂
9637:⋯
9619:∂
9604:∂
9583:∂
9568:∂
9545:∂
9530:∂
9522:⋯
9504:∂
9489:∂
9468:∂
9453:∂
9424:∂
9414:∂
9387:∂
9372:∂
9364:⋯
9353:∂
9338:∂
9324:∂
9309:∂
9282:∂
9272:∂
9238:∂
9230:∂
9220:⋮
9200:∂
9192:∂
9169:∂
9161:∂
9132:∂
9124:∂
9053:⋯
9010:⋮
9005:⋱
9000:⋮
8995:⋮
8970:⋯
8915:⋯
8838:∂
8820:∂
8812:⋯
8801:∂
8783:∂
8769:∂
8751:∂
8741:⋮
8736:⋱
8731:⋮
8726:⋮
8713:∂
8695:∂
8687:⋯
8676:∂
8661:∂
8647:∂
8632:∂
8616:∂
8598:∂
8590:⋯
8579:∂
8564:∂
8550:∂
8535:∂
8508:∂
8498:∂
8434:∂
8426:∂
8418:⋯
8397:∂
8389:∂
8365:∂
8357:∂
8347:⋮
8342:⋱
8337:⋮
8332:⋮
8309:∂
8301:∂
8293:⋯
8275:∂
8267:∂
8246:∂
8238:∂
8212:∂
8204:∂
8196:⋯
8178:∂
8170:∂
8149:∂
8141:∂
8112:∂
8104:∂
8070:∂
8055:∂
8047:⋯
8029:∂
8014:∂
7993:∂
7978:∂
7968:⋮
7963:⋱
7958:⋮
7953:⋮
7933:∂
7918:∂
7910:⋯
7892:∂
7877:∂
7856:∂
7841:∂
7818:∂
7803:∂
7795:⋯
7777:∂
7762:∂
7741:∂
7726:∂
7697:∂
7687:∂
7660:∂
7645:∂
7635:⋮
7622:∂
7607:∂
7591:∂
7576:∂
7549:∂
7539:∂
7505:∂
7497:∂
7489:⋯
7471:∂
7463:∂
7442:∂
7434:∂
7405:∂
7397:∂
7325:∂
7315:∂
7281:∂
7271:∂
7227:∂
7219:∂
7134:∂
7124:∂
7080:∂
7070:∂
7029:∂
7021:∂
6938:∂
6928:∂
6889:∂
6879:∂
6846:∂
6838:∂
6692:∂
6682:∂
6651:∂
6643:∂
6626:∂
6616:∂
6601:∂
6591:∂
6574:∂
6566:∂
6523:∂
6515:∂
6475:∂
6467:∂
6432:∂
6422:∂
6379:∂
6371:∂
6336:∂
6326:∂
6281:∂
6271:∂
6215:∂
6205:∂
6174:∂
6166:∂
6130:∂
6120:∂
6083:∂
6073:∂
6037:∂
6029:∂
5997:∂
5987:∂
5956:∂
5948:∂
5911:∂
5901:∂
5869:∂
5859:∂
5828:∂
5820:∂
5783:∂
5773:∂
5737:∂
5727:∂
5696:∂
5688:∂
5636:∂
5626:∂
5586:∂
5576:∂
5538:∂
5530:∂
5453:∂
5443:∂
5390:∂
5380:∂
5244:∂
5234:∂
5154:∂
5146:∂
5135:
5115:∇
5049:∂
5030:∂
4999:∇
4930:∂
4922:∂
4914:⋯
4893:∂
4885:∂
4861:∂
4853:∂
4843:⋮
4838:⋱
4833:⋮
4828:⋮
4805:∂
4797:∂
4789:⋯
4771:∂
4763:∂
4742:∂
4734:∂
4708:∂
4700:∂
4692:⋯
4674:∂
4666:∂
4645:∂
4637:∂
4612:∂
4604:∂
4527:∂
4509:∂
4501:⋯
4490:∂
4472:∂
4458:∂
4440:∂
4430:⋮
4425:⋱
4420:⋮
4415:⋮
4402:∂
4384:∂
4376:⋯
4365:∂
4350:∂
4336:∂
4321:∂
4305:∂
4287:∂
4279:⋯
4268:∂
4253:∂
4239:∂
4224:∂
4201:∂
4191:∂
4088:∂
4078:∂
4018:, or the
3957:∂
3942:∂
3934:⋯
3916:∂
3901:∂
3880:∂
3865:∂
3855:⋮
3850:⋱
3845:⋮
3840:⋮
3820:∂
3805:∂
3797:⋯
3779:∂
3764:∂
3743:∂
3728:∂
3705:∂
3690:∂
3682:⋯
3664:∂
3649:∂
3628:∂
3613:∂
3588:∂
3578:∂
3520:⋯
3431:⋯
3334:∂
3326:∂
3306:∇
3277:⋅
3260:∇
3230:∇
3141:∂
3133:∂
3098:∂
3090:∂
3080:⋮
3060:∂
3052:∂
3032:∇
2953:∂
2945:∂
2936:⋯
2917:∂
2909:∂
2886:∂
2878:∂
2852:∂
2844:∂
2786:⋯
2658:∂
2648:∂
2552:⋮
2393:⋯
2248:with its
2163:transpose
2073:rows and
1930:∂
1922:∂
1883:∂
1873:∂
1841:∂
1833:∂
1798:∂
1788:∂
1758:∂
1748:∂
1718:∂
1710:∂
1600:∂
1592:∂
1569:∂
1561:∂
1538:∂
1530:∂
1491:∂
1483:∂
1466:∇
1414:≤
1408:≤
1349:^
1307:^
1284:∂
1276:∂
1258:^
1235:∂
1227:∂
1209:^
1186:∂
1178:∂
1166:∇
1069:section.
1012:variables
886:Malliavin
773:Geometric
672:Laplacian
622:Dirichlet
533:Geometric
118:−
65:∫
36773:Manifold
36506:Integral
36449:Infinite
36444:Harmonic
36429:Binomial
36275:Gradient
36218:Volumes
36029:Quotient
35970:Notation
35801:Calculus
35728:Archived
35656:Software
35437:(2007).
35139:Jacobian
34609:, where
34519:See also
33199:′
30994:′
29891:i.e. by
29875:i.e. by
29327:′
29300:′
25632:cofactor
25559:cofactor
23450:′
23405:′
15506:‖
15498:‖
12430:i.e. by
12419:i.e. by
10225:sum rule
6809:Notation
6803:Notation
6797:Notation
6532:′
6046:′
5936:gradient
5808:gradient
5676:gradient
5645:′
5516:gradient
5511:Jacobian
5507:gradient
5282:of size
3182:gradient
3006: :
2999:gradient
2710:position
2698:velocity
2031:Notation
1087:gradient
1080:calculus
1008:function
1000:matrices
926:Glossary
836:Advanced
814:Jacobian
768:Exterior
698:Gradient
690:Theorems
657:Gradient
596:Integral
558:Binomial
543:Harmonic
408:improper
404:Integral
361:Integral
343:Reynolds
318:Quotient
247:Concepts
83:′
50:Calculus
36710:inverse
36698:inverse
36624:Fluxion
36434:Fourier
36300:Stokes'
36295:Green's
36017:Product
35877:Tangent
35380:2531677
35131:Hessian
33398:is the
30885:′
30871:′
29213:′
29199:′
23315:′
23301:′
23246:is any
7363:matrix
7258:matrix
7186:Matrix
7181:matrix
7111:matrix
6967:matrix
6866:Scalar
6817:Scalar
6776:Matrix
6752:Scalar
5198:of the
4580:matrix
3184:of the
2708:of the
2694:Example
2433:, by a
2195:is the
2173:is the
2155:, etc.
1909:Matrix
1820:Vector
1697:Scalar
1692:Matrix
1689:Vector
1686:Scalar
1040:physics
1020:vectors
921:History
819:Hessian
708:Stokes'
703:Green's
535: (
462: (
406: (
328:Inverse
303:Product
214: (
36793:Tensor
36715:Secant
36481:Abel's
36464:Taylor
36355:Matrix
36305:Gauss'
35887:Limits
35867:Secant
35857:Radian
35616:
35599:
35589:
35561:
35495:
35470:
35445:
35416:
35378:
35368:
34631:Here,
34579:Here,
33761:where
33572:, and
14352:, the
10388:, and
10339:, and
10327:, and
10278:, and
10266:, and
7193:(size
6995:(size
6782:(size
6766:(size
5076:, the
2997:, the
2730:scalar
2435:scalar
2343:vector
2193:|
2187:|
2177:, and
1997:Usages
1965:tensor
1683:Types
1334:where
1047:scalar
763:Tensor
758:Matrix
645:Vector
563:Taylor
521:Series
158:Limits
36657:Lists
36516:Ratio
36454:Power
36190:Euler
36007:Chain
35997:Power
35872:Slope
35678:SymPy
35559:S2CID
35317:(PDF)
35191:(PDF)
35113:(PDF)
34562:Notes
33818:with
26434:(see
26270:(see
18794:trace
10416:, or
10355:, or
10294:, or
10233:trace
7163:Size-
7051:Size-
6975:Size-
6909:Size-
6812:Type
6806:Type
6800:Type
4974:trace
4590:) by
4177:) by
3564:) as
2830:) as
2728:of a
2446:) as
2341:of a
2307:(1,1)
2175:trace
2135:(1,1)
1089:from
1073:Scope
586:Ratio
553:Power
472:Euler
450:Discs
445:Parts
313:Power
308:Chain
237:total
36526:Term
36521:Root
36260:Curl
35614:ISBN
35597:OCLC
35587:ISBN
35493:ISBN
35468:ISBN
35443:ISBN
35414:ISBN
35366:ISBN
35337:See
35325:2016
35203:2016
35125:2016
35095:2016
33402:and
30840:cos(
30832:sin(
29897:and
29881:and
29168:cos(
29160:sin(
23266:cos(
23258:sin(
22532:and
22220:and
22054:and
21901:and
18787:and
18723:and
18669:NOTE
10509:and
10495:and
10400:and
10223:and
6671:and
6456:and
6360:and
6254:and
6242:and
6062:and
5491:and
5343:and
4139:Note
4131:and
3191:The
2724:The
2337:The
2319:NOTE
2244:The
2212:NOTE
2179:det(
2056:real
1061:and
1030:and
667:Curl
627:Abel
591:Root
36002:Sum
35637:doi
35633:101
35551:doi
35526:doi
35522:188
35358:doi
35261:in
35255:by
35213:in
35137:to
33654:is
33522:of
33047:is
32967:is
32759:log
31404:adj
30848:ln(
29176:ln(
25454:cos
25416:cos
25362:sin
23284:);
23274:ln(
15283:is
14990:is
10229:not
5563:n×m
4039:in
3996:In
2993:In
2609:In
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2287:,1)
2185:or
2167:tr(
2106:,1)
986:In
298:Sum
34:or
36842::
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30614:tr
30543:=
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30093:ln
30068:=
29998:tr
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18410:=
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33920:d
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33869:P
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33770:X
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33732:i
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33721:f
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33705:)
33701:X
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33665:f
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33638:X
33634:(
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33601:=
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33475:1
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32919:X
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32839:1
32832:)
32827:I
32822:z
32819:+
32815:X
32811:(
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32775:=
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32766:X
32755:(
32751:d
32726:a
32723:d
32716:X
32712:)
32709:a
32703:1
32700:(
32696:e
32692:)
32688:X
32684:d
32681:(
32675:X
32671:a
32667:e
32661:1
32656:0
32630:=
32626:)
32620:X
32615:e
32611:(
32607:d
32580:1
32574:i
32568:n
32563:X
32558:)
32554:X
32550:d
32547:(
32542:i
32537:X
32530:1
32524:n
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32516:=
32513:i
32487:=
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32478:n
32473:X
32468:(
32464:d
32452:n
32432:H
32427:)
32422:X
32418:d
32415:(
32394:=
32390:)
32384:H
32378:X
32373:(
32369:d
32359:)
32355:(
32337:1
32329:X
32323:)
32318:X
32314:d
32310:(
32304:1
32296:X
32270:=
32266:)
32261:1
32253:X
32248:(
32244:d
32213:)
32208:X
32204:d
32201:(
32180:=
32176:)
32166:X
32161:(
32157:d
32132:)
32128:Y
32124:d
32121:(
32114:X
32110:+
32106:Y
32099:)
32095:X
32091:d
32088:(
32067:=
32064:)
32060:Y
32052:X
32048:(
32045:d
32035:)
32031:(
32015:)
32011:Y
32007:d
32004:(
31997:X
31993:+
31989:Y
31982:)
31978:X
31974:d
31971:(
31950:=
31947:)
31943:Y
31935:X
31931:(
31928:d
31918:)
31914:(
31898:)
31894:Y
31890:d
31887:(
31883:X
31879:+
31875:Y
31871:)
31867:X
31863:d
31860:(
31839:=
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31832:Y
31827:X
31823:(
31820:d
31794:Y
31790:d
31787:+
31783:X
31779:d
31758:=
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31751:Y
31747:+
31743:X
31739:(
31736:d
31710:X
31706:d
31702:a
31681:=
31678:)
31674:X
31670:a
31667:(
31664:d
31652:X
31647:a
31630:0
31609:=
31606:)
31602:A
31598:(
31595:d
31583:X
31578:A
31538:)
31533:X
31529:d
31524:1
31516:X
31510:(
31482:=
31479:)
31475:|
31470:X
31465:|
31455:(
31452:d
31429:)
31425:X
31421:d
31418:)
31414:X
31410:(
31401:(
31392:=
31388:)
31383:X
31379:d
31374:1
31366:X
31360:(
31349:|
31344:X
31339:|
31317:=
31314:)
31310:|
31305:X
31300:|
31296:(
31293:d
31270:)
31266:X
31262:d
31259:(
31232:=
31229:)
31226:)
31222:X
31218:(
31209:(
31206:d
31177:a
31149:)
31142:A
31138:x
31134:e
31129:A
31124:(
31096:=
31090:x
31081:)
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31071:x
31067:e
31063:(
31038:x
31032:A
31013:)
31009:)
31005:A
31001:x
30998:(
30990:g
30984:A
30979:(
30951:=
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30937:)
30934:)
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30926:x
30923:(
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30915:(
30891:)
30889:X
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30882:g
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30861:x
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30842:X
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30804:A
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30752:)
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30736:)
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30710:(
30681:)
30674:x
30665:U
30651:U
30642:)
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30634:(
30631:g
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30593:=
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30551:)
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30541:U
30522:]
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30512:2
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30500:x
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30479:1
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30327:(
30316:[
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30307:U
30303:|
30281:=
30273:2
30269:x
30259:|
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30243:2
30224:)
30222:x
30220:(
30218:U
30214:U
30195:)
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30153:(
30125:=
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30110:|
30105:U
30100:|
30076:)
30074:x
30072:(
30070:U
30066:U
30047:)
30040:x
30031:U
30019:1
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30005:(
29994:|
29989:U
29984:|
29962:=
29956:x
29947:|
29942:U
29937:|
29919:)
29917:x
29915:(
29913:U
29909:U
29900:X
29894:Y
29884:X
29878:Y
29834:v
29824:x
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29805:+
29799:x
29790:v
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29754:=
29748:x
29740:)
29736:v
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29724:(
29707:)
29705:x
29703:(
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29697:v
29692:)
29690:x
29688:(
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29582:x
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29570:u
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29563:g
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29544:x
29542:(
29540:u
29536:u
29475:A
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29464:x
29460:e
29456:=
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29437:A
29415:=
29409:x
29398:A
29394:x
29390:e
29371:x
29365:A
29346:A
29342:)
29338:A
29334:x
29331:(
29323:g
29318:=
29315:)
29311:A
29307:x
29304:(
29296:g
29290:A
29268:=
29262:x
29254:)
29250:A
29246:x
29243:(
29239:g
29219:)
29217:X
29215:(
29210:g
29205:)
29203:x
29201:(
29196:g
29191:)
29189:x
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29185:g
29180:)
29178:X
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29164:)
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29155:e
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29144:g
29138:x
29132:A
29112:1
29104:U
29098:)
29091:x
29082:U
29070:1
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29045:U
29035:+
29029:y
29023:x
29014:U
29008:2
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28970:1
28962:U
28954:x
28945:U
28934:(
28928:1
28920:U
28897:=
28891:y
28885:x
28875:1
28867:U
28860:2
28841:)
28839:y
28837:,
28835:x
28833:(
28831:U
28827:U
28807:1
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28762:U
28736:=
28730:x
28720:1
28712:U
28693:)
28691:x
28689:(
28687:U
28683:U
28664:V
28654:x
28645:U
28635:+
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28606:U
28584:=
28578:x
28570:)
28566:V
28558:U
28554:(
28537:)
28535:x
28533:(
28531:V
28527:V
28522:)
28520:x
28518:(
28516:U
28512:U
28493:V
28483:x
28474:U
28464:+
28458:x
28449:V
28435:U
28413:=
28407:x
28399:)
28395:V
28387:U
28383:(
28366:)
28364:x
28362:(
28360:V
28356:V
28351:)
28349:x
28347:(
28345:U
28341:U
28322:V
28315:x
28306:U
28296:+
28290:x
28281:V
28270:U
28248:=
28242:x
28234:)
28230:V
28225:U
28221:(
28204:)
28202:x
28200:(
28198:V
28194:V
28189:)
28187:x
28185:(
28183:U
28179:U
28158:x
28149:V
28139:+
28133:x
28124:U
28096:=
28090:x
28082:)
28078:V
28074:+
28070:U
28066:(
28049:)
28047:x
28045:(
28043:V
28039:V
28034:)
28032:x
28030:(
28028:U
28024:U
28005:B
27998:x
27989:U
27978:A
27956:=
27950:x
27941:B
27938:U
27935:A
27917:)
27915:x
27913:(
27911:U
27907:U
27903:,
27901:x
27896:B
27890:A
27869:x
27860:U
27850:a
27829:=
27823:x
27814:U
27810:a
27793:)
27791:x
27789:(
27787:U
27783:U
27774:Y
27746:x
27737:Y
27689:)
27682:1
27674:)
27670:X
27661:A
27651:X
27646:(
27640:X
27631:A
27619:+
27614:1
27606:)
27602:X
27599:A
27590:X
27585:(
27579:X
27576:A
27568:(
27562:|
27557:X
27552:A
27541:X
27535:|
27503:)
27491:A
27481:X
27474:1
27466:)
27462:X
27453:A
27443:X
27438:(
27426:+
27422:A
27413:X
27406:1
27398:)
27394:X
27391:A
27382:X
27377:(
27368:(
27362:|
27357:X
27352:A
27341:X
27335:|
27309:=
27302:X
27292:|
27287:X
27282:A
27271:X
27265:|
27245:A
27238:X
27234:,
27231:X
27225:A
27205:1
27197:)
27193:X
27190:A
27181:X
27176:(
27170:X
27167:A
27162:|
27157:X
27152:A
27141:X
27135:|
27131:2
27103:A
27093:X
27086:1
27078:)
27074:X
27065:A
27055:X
27050:(
27044:|
27039:X
27034:A
27023:X
27017:|
27013:2
26992:=
26985:X
26975:|
26970:X
26965:A
26954:X
26948:|
26928:A
26921:X
26917:,
26914:X
26908:A
26883:)
26878:1
26870:X
26865:(
26859:|
26854:X
26849:A
26838:X
26832:|
26828:2
26805:1
26797:X
26791:|
26786:X
26781:|
26776:|
26771:A
26766:|
26761:|
26751:X
26746:|
26742:2
26739:=
26734:1
26726:X
26720:|
26715:X
26710:A
26699:X
26693:|
26689:2
26668:=
26661:X
26651:|
26646:X
26641:A
26630:X
26624:|
26604:X
26600:,
26597:X
26591:A
26566:X
26561:2
26536:X
26532:2
26508:=
26500:+
26495:X
26484:|
26479:X
26468:X
26462:|
26438:)
26411:)
26406:+
26401:X
26396:(
26391:2
26368:+
26363:X
26358:2
26337:=
26330:X
26320:|
26315:X
26304:X
26298:|
26274:)
26247:)
26242:1
26234:X
26229:(
26223:|
26218:n
26213:X
26208:|
26204:n
26181:1
26173:X
26167:|
26162:n
26157:X
26152:|
26148:n
26127:=
26120:X
26110:|
26105:n
26100:X
26095:|
26075:n
26051:)
26046:1
26038:X
26033:(
26027:|
26022:B
26019:X
26016:A
26011:|
25987:1
25979:X
25973:|
25968:B
25965:X
25962:A
25957:|
25935:=
25928:X
25918:|
25913:B
25910:X
25907:A
25902:|
25882:X
25876:B
25870:A
25845:)
25840:1
25832:X
25827:(
25802:1
25794:X
25771:=
25764:X
25754:|
25749:X
25745:a
25741:|
25716:X
25710:a
25685:)
25680:1
25672:X
25667:(
25661:|
25656:X
25651:|
25647:=
25644:)
25641:X
25638:(
25609:1
25601:X
25595:|
25590:X
25585:|
25581:=
25572:)
25568:X
25565:(
25538:=
25531:X
25521:|
25516:X
25511:|
25472:)
25468:)
25464:X
25460:(
25451:(
25430:)
25426:X
25422:(
25395:=
25388:X
25379:)
25376:)
25372:X
25368:(
25359:(
25314:)
25308:X
25303:e
25299:(
25273:X
25268:e
25246:=
25239:X
25229:)
25223:X
25218:e
25214:(
25168:)
25162:1
25156:i
25150:n
25145:X
25139:A
25133:i
25128:X
25122:(
25115:1
25109:n
25104:0
25101:=
25098:i
25070:1
25064:i
25058:n
25053:X
25047:A
25041:i
25036:X
25029:1
25023:n
25018:0
25015:=
25012:i
24986:=
24979:X
24969:)
24963:n
24958:X
24952:A
24947:(
24921:n
24918:,
24915:X
24909:A
24884:)
24879:1
24873:n
24868:X
24863:(
24858:n
24835:1
24829:n
24824:X
24819:n
24798:=
24791:X
24781:)
24776:n
24771:X
24766:(
24740:n
24722:B
24719:X
24716:A
24713:C
24709:+
24699:B
24695:X
24686:C
24676:A
24646:C
24636:A
24626:X
24616:B
24611:+
24607:A
24604:C
24595:X
24591:B
24569:=
24562:X
24552:)
24548:C
24539:X
24535:B
24532:X
24529:A
24525:(
24500:X
24494:C
24488:B
24482:A
24457:B
24447:A
24423:A
24420:B
24398:=
24391:X
24382:)
24378:X
24375:A
24372:B
24368:(
24353:=
24346:X
24337:)
24333:B
24330:X
24327:A
24323:(
24299:X
24293:B
24287:A
24262:)
24257:1
24249:X
24244:(
24232:A
24220:)
24215:1
24207:X
24202:(
24174:1
24166:X
24160:A
24154:1
24146:X
24120:=
24113:X
24104:)
24100:A
24095:1
24088:X
24083:(
24058:X
24052:A
24033:X
24028:)
24017:A
24012:+
24008:A
24003:(
23980:)
23969:A
23964:+
23960:A
23955:(
23944:X
23921:=
23914:X
23904:)
23900:X
23897:A
23888:X
23883:(
23857:X
23851:A
23832:A
23803:A
23780:=
23773:X
23763:)
23759:A
23750:X
23745:(
23729:=
23722:X
23712:)
23702:X
23698:A
23694:(
23668:X
23662:A
23637:A
23613:A
23591:=
23584:X
23575:)
23571:A
23568:X
23564:(
23549:=
23542:X
23533:)
23529:X
23526:A
23522:(
23497:X
23491:A
23466:)
23462:)
23458:X
23454:(
23446:g
23440:(
23417:)
23413:X
23409:(
23401:g
23378:=
23371:X
23362:)
23358:)
23355:X
23352:(
23349:g
23345:(
23321:)
23319:X
23317:(
23312:g
23307:)
23305:x
23303:(
23298:g
23293:)
23291:x
23289:(
23287:g
23278:)
23276:X
23270:)
23268:X
23262:)
23260:X
23253:e
23244:)
23242:X
23240:(
23238:g
23216:X
23207:)
23203:U
23199:(
23184:a
23163:=
23156:X
23147:)
23143:U
23139:a
23136:(
23113:)
23111:X
23109:(
23107:U
23103:U
23099:,
23096:X
23090:a
23068:X
23059:)
23055:V
23051:(
23036:+
23029:X
23020:)
23016:U
23012:(
22979:=
22972:X
22963:)
22959:V
22955:+
22951:U
22947:(
22924:)
22922:X
22920:(
22918:V
22914:V
22909:)
22907:X
22905:(
22903:U
22899:U
22880:I
22858:=
22851:X
22842:)
22838:X
22834:(
22790:b
22784:a
22779:X
22768:C
22763:+
22753:a
22747:b
22742:X
22737:C
22708:)
22697:b
22691:a
22686:X
22675:C
22670:+
22660:a
22654:b
22649:X
22644:C
22639:(
22616:=
22609:X
22600:)
22596:b
22591:X
22587:(
22583:C
22573:)
22568:a
22563:X
22559:(
22541:X
22535:C
22529:b
22523:a
22498:a
22493:)
22489:b
22485:+
22481:a
22476:X
22472:(
22468:)
22457:C
22452:+
22448:C
22443:(
22414:)
22403:a
22398:)
22394:b
22390:+
22386:a
22381:X
22377:(
22373:)
22362:C
22357:+
22353:C
22348:(
22343:(
22320:=
22313:X
22304:)
22300:b
22296:+
22292:a
22287:X
22283:(
22279:C
22269:)
22264:b
22260:+
22256:a
22251:X
22247:(
22229:X
22223:C
22217:b
22211:a
22186:a
22180:b
22151:b
22145:a
22123:=
22116:X
22106:b
22095:X
22083:a
22063:X
22057:b
22051:a
22026:b
22020:a
21991:a
21985:b
21963:=
21956:X
21946:b
21941:X
21930:a
21910:X
21904:b
21898:a
21887:X
21870:,
21862:j
21859:i
21855:X
21845:U
21809:)
21800:j
21797:i
21793:X
21783:U
21766:)
21759:U
21750:)
21746:U
21742:(
21739:g
21730:(
21724:(
21695:)
21686:j
21683:i
21679:X
21669:U
21655:U
21646:)
21642:U
21638:(
21635:g
21625:(
21597:=
21589:j
21586:i
21582:X
21573:)
21569:U
21565:(
21562:g
21544:)
21542:X
21540:(
21538:U
21534:U
21512:X
21503:u
21491:u
21483:)
21480:u
21477:(
21474:g
21462:g
21454:)
21451:g
21448:(
21445:f
21418:=
21411:X
21402:)
21399:)
21396:u
21393:(
21390:g
21387:(
21384:f
21367:)
21365:X
21363:(
21361:u
21357:u
21335:X
21326:u
21314:u
21306:)
21303:u
21300:(
21297:g
21270:=
21263:X
21254:)
21251:u
21248:(
21245:g
21228:)
21226:X
21224:(
21222:u
21218:u
21196:X
21187:u
21178:v
21175:+
21168:X
21159:v
21150:u
21129:=
21122:X
21113:v
21110:u
21093:)
21091:X
21089:(
21087:v
21083:v
21078:)
21076:X
21074:(
21072:u
21068:u
21046:X
21037:v
21028:+
21021:X
21012:u
20985:=
20978:X
20969:)
20966:v
20963:+
20960:u
20957:(
20940:)
20938:X
20936:(
20934:v
20930:v
20925:)
20923:X
20921:(
20919:u
20915:u
20893:X
20884:u
20875:a
20854:=
20847:X
20838:u
20835:a
20818:)
20816:X
20814:(
20812:u
20808:u
20802:X
20796:a
20777:0
20748:0
20725:=
20718:X
20709:a
20691:X
20685:a
20676:X
20668:X
20639:X
20630:y
20590:.
20580:B
20576:X
20567:C
20557:A
20552:+
20548:B
20545:X
20542:A
20539:C
20535:=
20528:X
20518:)
20514:C
20505:X
20501:B
20498:X
20495:A
20491:(
20456:.
20452:A
20449:C
20440:X
20436:B
20432:+
20422:C
20412:A
20402:X
20392:B
20387:=
20380:X
20370:)
20366:C
20357:X
20353:B
20350:X
20347:A
20343:(
20300:)
20295:X
20291:d
20281:)
20270:B
20266:X
20257:C
20247:A
20242:+
20238:B
20235:X
20232:A
20229:C
20224:(
20218:(
20208:=
20197:)
20192:X
20188:d
20184:)
20179:A
20176:C
20165:X
20161:B
20157:+
20147:C
20137:A
20127:X
20117:B
20111:(
20106:(
20096:=
20085:)
20081:)
20077:X
20073:d
20070:(
20066:A
20063:C
20052:X
20048:B
20043:(
20033:+
20029:)
20025:)
20021:X
20017:d
20014:(
20004:C
19994:A
19984:X
19974:B
19968:(
19958:=
19947:)
19936:X
19932:B
19928:)
19924:X
19920:d
19917:(
19913:A
19910:C
19905:(
19895:+
19891:)
19880:C
19870:A
19860:X
19850:B
19845:)
19841:X
19837:d
19834:(
19830:(
19820:=
19809:)
19798:X
19794:B
19790:)
19786:X
19782:d
19779:(
19775:A
19772:C
19767:(
19757:+
19753:)
19743:)
19733:)
19728:X
19724:d
19721:(
19717:B
19714:X
19711:A
19708:C
19703:(
19698:(
19688:=
19677:)
19666:X
19662:B
19658:)
19654:X
19650:d
19646:(
19641:A
19638:C
19634:(
19625:+
19621:)
19611:)
19606:X
19602:d
19599:(
19595:B
19592:X
19589:A
19586:C
19581:(
19571:=
19560:)
19549:X
19545:B
19541:)
19537:X
19533:d
19530:(
19526:A
19523:C
19518:(
19508:+
19504:)
19499:)
19489:X
19484:(
19480:d
19476:B
19473:X
19470:A
19467:C
19462:(
19452:=
19441:)
19430:X
19426:B
19422:)
19418:X
19415:A
19412:C
19408:(
19405:d
19401:(
19391:+
19387:)
19382:)
19372:X
19368:B
19364:(
19360:d
19356:X
19353:A
19350:C
19345:(
19335:=
19324:)
19313:X
19309:B
19305:)
19301:X
19298:A
19295:C
19290:(
19286:d
19283:+
19279:)
19268:X
19264:B
19260:(
19257:d
19253:X
19250:A
19247:C
19242:(
19232:=
19221:)
19216:)
19206:X
19202:B
19199:X
19196:A
19193:C
19189:(
19185:d
19181:(
19171:=
19167:)
19157:X
19153:B
19150:X
19147:A
19144:C
19140:(
19130:d
19127:=
19120:)
19116:C
19107:X
19103:B
19100:X
19097:A
19093:(
19084:d
19061::
19054:X
19045:)
19041:C
19032:X
19028:B
19025:X
19022:A
19018:(
18979:)
18975:C
18972:B
18969:A
18966:D
18962:(
18953:=
18950:)
18946:B
18943:A
18940:D
18937:C
18933:(
18924:=
18921:)
18917:A
18914:D
18911:C
18908:B
18904:(
18895:=
18888:)
18884:D
18881:C
18878:B
18875:A
18871:(
18857:)
18847:A
18842:(
18832:=
18825:)
18821:A
18817:(
18759:g
18750:)
18746:g
18742:(
18738:f
18707:u
18698:)
18694:u
18690:(
18686:g
18642:U
18631:x
18622:v
18612:+
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18602:x
18593:U
18583:(
18569:v
18543:x
18534:v
18520:U
18516:+
18512:v
18502:x
18493:U
18465:=
18459:x
18451:)
18447:v
18439:U
18435:(
18418:)
18416:x
18414:(
18412:v
18408:v
18403:)
18401:x
18399:(
18397:U
18393:U
18366:g
18357:)
18353:g
18349:(
18345:f
18331:u
18322:)
18318:u
18314:(
18310:g
18297:x
18288:u
18257:x
18248:u
18234:u
18225:)
18221:u
18217:(
18213:g
18199:g
18190:)
18186:g
18182:(
18178:f
18150:=
18144:x
18136:)
18133:)
18129:u
18125:(
18121:g
18117:(
18113:f
18095:)
18093:x
18091:(
18089:u
18085:u
18058:u
18049:)
18045:u
18041:(
18037:g
18024:x
18015:u
17984:x
17975:u
17961:u
17952:)
17948:u
17944:(
17940:g
17912:=
17906:x
17898:)
17894:u
17890:(
17886:g
17868:)
17866:x
17864:(
17862:u
17858:u
17833:)
17827:x
17818:v
17808:(
17793:u
17788:+
17784:v
17774:x
17765:u
17734:x
17725:v
17705:u
17700:+
17696:v
17682:)
17676:x
17667:u
17657:(
17634:=
17628:x
17620:)
17616:v
17602:u
17597:(
17580:)
17578:x
17576:(
17574:v
17570:v
17565:)
17563:x
17561:(
17559:u
17555:u
17534:x
17525:v
17515:+
17509:x
17500:u
17472:=
17466:x
17458:)
17454:v
17450:+
17446:u
17442:(
17425:)
17423:x
17421:(
17419:v
17415:v
17410:)
17408:x
17406:(
17404:u
17400:u
17375:)
17369:x
17360:u
17350:(
17327:=
17321:x
17306:u
17287:)
17285:x
17283:(
17281:u
17277:u
17252:A
17244:x
17235:u
17204:x
17195:u
17184:A
17162:=
17156:x
17147:u
17142:A
17124:)
17122:x
17120:(
17118:u
17114:u
17110:,
17108:x
17103:A
17082:x
17073:u
17063:a
17042:=
17036:x
17027:u
17023:a
17006:)
17004:x
17002:(
17000:u
16996:u
16992:,
16989:x
16983:a
16964:0
16942:=
16936:x
16927:a
16908:x
16902:a
16893:,
16890:y
16882:,
16879:y
16851:x
16842:y
16793:a
16785:x
16775:a
16767:x
16735:a
16727:x
16712:)
16707:a
16699:x
16695:(
16671:=
16664:x
16650:a
16642:x
16619:x
16613:a
16595:)
16590:e
16585:+
16580:x
16573:D
16568:(
16563:C
16550:A
16544:+
16541:)
16536:b
16531:+
16526:x
16519:A
16514:(
16503:C
16489:D
16463:D
16456:C
16445:)
16439:b
16434:+
16429:x
16422:A
16417:(
16414:+
16409:A
16396:C
16384:)
16378:e
16373:+
16368:x
16361:D
16356:(
16335:=
16327:x
16316:)
16311:e
16306:+
16301:x
16294:D
16289:(
16284:C
16273:)
16267:b
16262:+
16257:x
16250:A
16245:(
16226:x
16220:e
16214:D
16208:C
16202:b
16196:A
16176:x
16170:)
16158:a
16150:b
16145:+
16134:b
16126:a
16120:(
16097:)
16085:a
16077:b
16072:+
16061:b
16053:a
16047:(
16035:x
16011:=
16003:x
15990:b
15977:x
15969:x
15956:a
15934:x
15928:b
15922:a
15898:x
15888:u
15860:a
15852:x
15842:u
15808:x
15798:u
15767:x
15757:u
15740:a
15717:=
15710:x
15700:u
15689:a
15678:=
15671:x
15662:)
15658:u
15650:a
15646:(
15629:)
15627:x
15625:(
15623:u
15619:u
15612:x
15606:a
15587:x
15583:2
15555:x
15550:2
15529:=
15522:x
15511:2
15502:x
15487:=
15480:x
15470:x
15459:x
15448:=
15441:x
15432:)
15428:x
15420:x
15416:(
15384:A
15380:2
15359:=
15346:x
15337:x
15327:x
15322:A
15311:x
15304:2
15280:A
15274:x
15268:A
15243:A
15238:+
15234:A
15212:=
15199:x
15190:x
15180:x
15175:A
15164:x
15157:2
15137:x
15131:A
15112:x
15107:A
15103:2
15081:A
15070:x
15065:2
15044:=
15037:x
15027:x
15022:A
15011:x
14987:A
14981:x
14975:A
14956:x
14951:)
14940:A
14935:+
14931:A
14926:(
14903:)
14892:A
14887:+
14883:A
14878:(
14867:x
14844:=
14837:x
14827:x
14822:A
14811:x
14791:x
14785:A
14766:b
14755:A
14731:A
14720:b
14697:=
14690:x
14680:x
14675:A
14664:b
14644:x
14638:b
14632:x
14626:A
14607:a
14578:a
14555:=
14548:x
14538:a
14527:x
14516:=
14509:x
14499:x
14488:a
14459:=
14452:x
14443:)
14439:a
14431:x
14427:(
14418:=
14411:x
14402:)
14398:x
14390:a
14386:(
14368:x
14362:a
14339:H
14310:H
14287:=
14274:x
14265:x
14256:f
14251:2
14212:x
14202:v
14192:,
14185:x
14175:u
14147:u
14136:A
14127:x
14117:v
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14098:A
14090:x
14080:u
14046:x
14036:v
14026:,
14019:x
14009:u
13978:x
13968:u
13951:A
13939:v
13934:+
13927:x
13917:v
13906:A
13895:u
13872:=
13865:x
13855:v
13850:A
13839:u
13828:=
13821:x
13812:)
13808:v
13803:A
13795:u
13791:(
13773:x
13767:A
13763:,
13761:)
13759:x
13757:(
13755:v
13751:v
13746:)
13744:x
13742:(
13740:u
13736:u
13712:x
13702:v
13692:,
13685:x
13675:u
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13639:x
13629:v
13619:+
13615:v
13607:x
13597:u
13563:x
13553:v
13543:,
13536:x
13526:u
13495:x
13485:u
13468:v
13463:+
13456:x
13446:v
13429:u
13406:=
13399:x
13389:v
13378:u
13367:=
13360:x
13351:)
13347:v
13339:u
13335:(
13318:)
13316:x
13314:(
13312:v
13308:v
13303:)
13301:x
13299:(
13297:u
13293:u
13271:x
13262:u
13250:u
13242:)
13239:u
13236:(
13233:g
13221:g
13213:)
13210:g
13207:(
13204:f
13177:=
13170:x
13161:)
13158:)
13155:u
13152:(
13149:g
13146:(
13143:f
13126:)
13124:x
13122:(
13120:u
13116:u
13094:x
13085:u
13073:u
13065:)
13062:u
13059:(
13056:g
13029:=
13022:x
13013:)
13010:u
13007:(
13004:g
12987:)
12985:x
12983:(
12981:u
12977:u
12955:x
12946:u
12937:v
12934:+
12927:x
12918:v
12909:u
12888:=
12881:x
12872:v
12869:u
12852:)
12850:x
12848:(
12846:v
12842:v
12837:)
12835:x
12833:(
12831:u
12827:u
12805:x
12796:v
12787:+
12780:x
12771:u
12744:=
12737:x
12728:)
12725:v
12722:+
12719:u
12716:(
12699:)
12697:x
12695:(
12693:v
12689:v
12684:)
12682:x
12680:(
12678:u
12674:u
12652:x
12643:u
12634:a
12613:=
12606:x
12597:u
12594:a
12577:)
12575:x
12573:(
12571:u
12567:u
12560:x
12554:a
12535:0
12506:0
12483:=
12476:x
12467:a
12449:x
12443:a
12433:x
12422:x
12395:y
12389:x
12380:=
12373:x
12364:y
12316:g
12307:)
12303:g
12299:(
12295:f
12281:u
12272:)
12268:u
12264:(
12260:g
12246:x
12236:u
12204:x
12194:u
12180:u
12171:)
12167:u
12163:(
12159:g
12145:g
12136:)
12132:g
12128:(
12124:f
12096:=
12089:x
12080:)
12077:)
12073:u
12069:(
12065:g
12061:(
12057:f
12039:)
12037:x
12035:(
12033:u
12029:u
12007:u
11998:)
11994:u
11990:(
11986:g
11972:x
11962:u
11930:x
11920:u
11906:u
11897:)
11893:u
11889:(
11885:g
11857:=
11850:x
11841:)
11837:u
11833:(
11829:g
11811:)
11809:x
11807:(
11805:u
11801:u
11779:x
11769:v
11759:+
11752:x
11742:u
11714:=
11707:x
11698:)
11694:v
11690:+
11686:u
11682:(
11665:)
11663:x
11661:(
11659:v
11655:v
11650:)
11648:x
11646:(
11644:u
11640:u
11615:A
11606:x
11596:u
11564:x
11554:u
11543:A
11521:=
11514:x
11504:u
11499:A
11481:)
11479:x
11477:(
11475:u
11471:u
11467:,
11464:x
11458:A
11433:u
11424:x
11415:v
11406:+
11399:x
11389:u
11379:v
11354:x
11345:v
11335:u
11331:+
11324:x
11314:u
11304:v
11283:=
11276:x
11266:u
11262:v
11245:)
11243:x
11241:(
11239:u
11235:u
11230:)
11228:x
11226:(
11224:v
11220:v
11195:a
11186:x
11177:v
11146:x
11137:v
11127:a
11105:=
11098:x
11088:a
11084:v
11066:x
11060:a
11054:)
11052:x
11050:(
11048:v
11044:v
11022:x
11012:u
11002:a
10981:=
10974:x
10963:u
10959:a
10942:)
10940:x
10938:(
10936:u
10932:u
10928:,
10925:x
10919:a
10900:A
10871:A
10848:=
10841:x
10831:A
10820:x
10800:x
10794:A
10769:A
10745:A
10723:=
10716:x
10706:x
10701:A
10682:x
10676:A
10657:I
10635:=
10628:x
10618:x
10585:0
10563:=
10556:x
10546:a
10527:x
10521:a
10512:x
10506:y
10498:x
10492:y
10463:x
10453:y
10422:.
10419:X
10413:x
10408:x
10403:V
10397:U
10391:E
10385:D
10379:C
10373:B
10367:A
10361:;
10358:X
10352:x
10347:x
10342:v
10336:u
10330:e
10324:d
10318:c
10312:b
10306:a
10300:;
10297:X
10291:x
10286:x
10281:v
10275:u
10269:e
10263:d
10257:c
10251:b
10245:a
10188:.
10183:]
10172:q
10169:p
10165:x
10156:y
10135:2
10132:p
10128:x
10119:y
10103:1
10100:p
10096:x
10087:y
10047:q
10044:2
10040:x
10031:y
10006:x
9997:y
9977:x
9968:y
9950:q
9947:1
9943:x
9934:y
9909:x
9900:y
9880:x
9871:y
9859:[
9854:=
9843:X
9834:y
9821:.
9816:]
9805:n
9801:x
9790:m
9786:y
9764:n
9760:x
9749:2
9745:y
9728:n
9724:x
9713:1
9709:y
9668:2
9664:x
9653:m
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9627:2
9623:x
9612:2
9608:y
9591:2
9587:x
9576:1
9572:y
9553:1
9549:x
9538:m
9534:y
9512:1
9508:x
9497:2
9493:y
9476:1
9472:x
9461:1
9457:y
9444:[
9439:=
9428:x
9418:y
9404:.
9399:]
9390:x
9380:m
9376:y
9356:x
9346:2
9342:y
9327:x
9317:1
9313:y
9300:[
9295:=
9285:x
9276:y
9262:.
9257:]
9246:n
9242:x
9233:y
9208:2
9204:x
9195:y
9177:1
9173:x
9164:y
9152:[
9147:=
9136:x
9127:y
9082:.
9077:]
9069:n
9066:m
9062:x
9058:d
9046:2
9043:m
9039:x
9035:d
9028:1
9025:m
9021:x
9017:d
8986:n
8983:2
8979:x
8975:d
8959:x
8955:d
8944:x
8940:d
8931:n
8928:1
8924:x
8920:d
8904:x
8900:d
8889:x
8885:d
8879:[
8874:=
8866:X
8862:d
8855:.
8850:]
8841:x
8831:n
8828:m
8824:y
8804:x
8794:2
8791:m
8787:y
8772:x
8762:1
8759:m
8755:y
8716:x
8706:n
8703:2
8699:y
8679:x
8665:y
8650:x
8636:y
8619:x
8609:n
8606:1
8602:y
8582:x
8568:y
8553:x
8539:y
8526:[
8521:=
8511:x
8502:Y
8461:.
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7420:=
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5024:=
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4210:=
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4072:=
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3599:=
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1514:=
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1472:=
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112:b
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90:t
87:(
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69:a
38:.
20:)
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