9794:
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9789:{\displaystyle {\begin{bmatrix}a_{1,1}&a_{1,2}\\a_{2,1}&a_{2,2}\\\end{bmatrix}}\otimes {\begin{bmatrix}b_{1,1}&b_{1,2}\\b_{2,1}&b_{2,2}\\\end{bmatrix}}={\begin{bmatrix}a_{1,1}{\begin{bmatrix}b_{1,1}&b_{1,2}\\b_{2,1}&b_{2,2}\\\end{bmatrix}}&a_{1,2}{\begin{bmatrix}b_{1,1}&b_{1,2}\\b_{2,1}&b_{2,2}\\\end{bmatrix}}\\a_{2,1}{\begin{bmatrix}b_{1,1}&b_{1,2}\\b_{2,1}&b_{2,2}\\\end{bmatrix}}&a_{2,2}{\begin{bmatrix}b_{1,1}&b_{1,2}\\b_{2,1}&b_{2,2}\\\end{bmatrix}}\\\end{bmatrix}}={\begin{bmatrix}a_{1,1}b_{1,1}&a_{1,1}b_{1,2}&a_{1,2}b_{1,1}&a_{1,2}b_{1,2}\\a_{1,1}b_{2,1}&a_{1,1}b_{2,2}&a_{1,2}b_{2,1}&a_{1,2}b_{2,2}\\a_{2,1}b_{1,1}&a_{2,1}b_{1,2}&a_{2,2}b_{1,1}&a_{2,2}b_{1,2}\\a_{2,1}b_{2,1}&a_{2,1}b_{2,2}&a_{2,2}b_{2,1}&a_{2,2}b_{2,2}\\\end{bmatrix}}.}
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that satisfy the property are isomorphic through a unique isomorphism that is compatible with the universal property. When this definition is used, the other definitions may be viewed as constructions of objects satisfying the universal property and as proofs that there are objects satisfying the
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548:. There are several equivalent ways to define it. Most consist of defining explicitly a vector space that is called a tensor product, and, generally, the equivalence proof results almost immediately from the basic properties of the vector spaces that are so defined.
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19039:. It captures the algebraic essence of tensoring, without making any specific reference to what is being tensored. Thus, all tensor products can be expressed as an application of the monoidal category to some particular setting, acting on some particular objects.
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14664:{\displaystyle {\begin{aligned}&\forall a,a_{1},a_{2}\in A,\forall b,b_{1},b_{2}\in B,{\text{ for all }}r\in R:\\&(a_{1},b)+(a_{2},b)-(a_{1}+a_{2},b),\\&(a,b_{1})+(a,b_{2})-(a,b_{1}+b_{2}),\\&(ar,b)-(a,rb).\\\end{aligned}}}
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4065:. It follows that this is a (non-constructive) way to define the tensor product of two vector spaces. In this context, the preceding constructions of tensor products may be viewed as proofs of existence of the tensor product so defined.
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2769:{\displaystyle {\begin{aligned}x\otimes y&={\biggl (}\sum _{v\in B_{V}}x_{v}\,v{\biggr )}\otimes {\biggl (}\sum _{w\in B_{W}}y_{w}\,w{\biggr )}\\&=\sum _{v\in B_{V}}\sum _{w\in B_{W}}x_{v}y_{w}\,v\otimes w.\end{aligned}}}
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A consequence of this approach is that every property of the tensor product can be deduced from the universal property, and that, in practice, one may forget the method that has been used to prove its existence.
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A limitation of this definition of the tensor product is that, if one changes bases, a different tensor product is defined. However, the decomposition on one basis of the elements of the other basis defines a
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of the algebraic tensor product discussed above. However, it does not satisfy the obvious analogue of the universal property defining tensor products; the morphisms for that property must be restricted to
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between the two tensor products of vector spaces, which allows identifying them. Also, contrarily to the two following alternative definitions, this definition cannot be extended into a definition of the
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18124:{\displaystyle \psi _{i}(x_{1},\ldots ,x_{n})=\sum _{j_{2}=1}^{n}\sum _{j_{3}=1}^{n}\cdots \sum _{j_{d}=1}^{n}a_{ij_{2}j_{3}\cdots j_{d}}x_{j_{2}}x_{j_{3}}\cdots x_{j_{d}}\;\;{\mbox{for }}i=1,\ldots ,n}
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18471:{\displaystyle {\mbox{rank}}{\begin{pmatrix}x_{1}&x_{2}&\cdots &x_{n}\\\psi _{1}(\mathbf {x} )&\psi _{2}(\mathbf {x} )&\cdots &\psi _{n}(\mathbf {x} )\end{pmatrix}}\leq 1}
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19929:{\displaystyle \operatorname {Sym} ^{n}V:=\underbrace {V\otimes \dots \otimes V} _{n}{\big /}(\dots \otimes v_{i}\otimes v_{i+1}\otimes \dots -\dots \otimes v_{i+1}\otimes v_{i}\otimes \dots )}
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of subspaces. In such cases, the tensor product of two spaces can be decomposed into sums of products of the subspaces (in analogy to the way that multiplication distributes over addition).
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10110:{\displaystyle T_{s}^{r}(V)=\underbrace {V\otimes \cdots \otimes V} _{r}\otimes \underbrace {V^{*}\otimes \cdots \otimes V^{*}} _{s}=V^{\otimes r}\otimes \left(V^{*}\right)^{\otimes s}.}
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13916:{\displaystyle {\begin{aligned}\varphi (a+a',b)&=\varphi (a,b)+\varphi (a',b)\\\varphi (a,b+b')&=\varphi (a,b)+\varphi (a,b')\\\varphi (ar,b)&=\varphi (a,rb)\end{aligned}}}
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Like the universal property above, the following characterization may also be used to determine whether or not a given vector space and given bilinear map form a tensor product.
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8580:{\displaystyle A={\begin{bmatrix}a_{1,1}&a_{1,2}\\a_{2,1}&a_{2,2}\\\end{bmatrix}},\qquad B={\begin{bmatrix}b_{1,1}&b_{1,2}\\b_{2,1}&b_{2,2}\\\end{bmatrix}},}
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3716:{\displaystyle {\begin{aligned}(v_{1}+v_{2},w)&-(v_{1},w)-(v_{2},w),\\(v,w_{1}+w_{2})&-(v,w_{1})-(v,w_{2}),\\(sv,w)&-s(v,w),\\(v,sw)&-s(v,w),\end{aligned}}}
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788:. This definition can be formalized in the following way (this formalization is rarely used in practice, as the preceding informal definition is generally sufficient):
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However, these kinds of notation are not universally present in array languages. Other array languages may require explicit treatment of indices (for example,
14995:{\displaystyle {\begin{aligned}q(a_{1}+a_{2},b)&=q(a_{1},b)+q(a_{2},b),\\q(a,b_{1}+b_{2})&=q(a,b_{1})+q(a,b_{2}),\\q(ar,b)&=q(a,rb).\end{aligned}}}
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6809:{\displaystyle {\begin{alignedat}{4}\;&&X\times Y&&\;\to \;&Z\\&&(f,g)&&\;\mapsto \;&f\otimes g\\\end{alignedat}}}
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18561:. However, such a construction is no longer uniquely specified: in many cases, there are multiple natural topologies on the algebraic tensor product.
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In situations where the imposition of an inner product is inappropriate, one can still attempt to complete the algebraic tensor product, as a
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satisfied by the tensor product is described. As for every universal property, two objects that satisfy the property are related by a unique
19007:
It should be mentioned that, though called "tensor product", this is not a tensor product of graphs in the above sense; actually it is the
12239:
10897:
Thus, the components of the tensor product of two tensors are the ordinary product of the components of each tensor. Another example: let
19938:
That is, in the symmetric algebra two adjacent vectors (and therefore all of them) can be interchanged. The resulting objects are called
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The interplay of evaluation and coevaluation can be used to characterize finite-dimensional vector spaces without referring to bases.
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3095:. Therefore, the tensor product is a generalization of the outer product, that is, an abstraction of it beyond coordinate vectors.
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19346:{\displaystyle V\wedge V:=V\otimes V{\big /}{\bigl \{}v_{1}\otimes v_{2}+v_{2}\otimes v_{1}\mid (v_{1},v_{2})\in V^{2}{\bigr \}}.}
19762:{\displaystyle V\odot V:=V\otimes V{\big /}{\bigl \{}v_{1}\otimes v_{2}-v_{2}\otimes v_{1}\mid (v_{1},v_{2})\in V^{2}{\bigr \}}.}
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12955:{\displaystyle {\begin{cases}\mathrm {Hom} (U,V)\to U^{*}\otimes V\\F\mapsto \sum _{i}u_{i}^{*}\otimes F(u_{i}).\end{cases}}}
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The tensor product of two vector spaces captures the properties of all bilinear maps in the sense that a bilinear map from
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10888:{\displaystyle (F\otimes G)_{i_{1}i_{2}\cdots i_{m+n}}=F_{i_{1}i_{2}\cdots i_{m}}G_{i_{m+1}i_{m+2}i_{m+3}\cdots i_{m+n}}.}
11219:
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2180:{\displaystyle (v\otimes w)(x,y):=\sum _{v'\in B_{V}}\sum _{w'\in B_{W}}x_{v'}y_{w'}\,(v\otimes w)(v',w')=x_{v}\,y_{w}.}
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12175:
8252:, then the same is true for all above defined linear maps. In particular, the tensor product with a vector space is an
13678:
13473:
11611:{\displaystyle {\begin{cases}K\to V\otimes V^{*}\\\lambda \mapsto \sum _{i}\lambda v_{i}\otimes v_{i}^{*}\end{cases}}}
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24:
15706:, as was mentioned above. For modules over a general (commutative) ring, not every module is free. For example,
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The "universal-property definition" of the tensor product of two vector spaces is the following (recall that a
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16709:{\displaystyle (a_{1}\otimes b_{1})\cdot (a_{2}\otimes b_{2})=(a_{1}\cdot a_{2})\otimes (b_{1}\cdot b_{2}).}
16481:
measure the defect of the tensor product being not left exact. All higher Tor functors are assembled in the
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11098:{\displaystyle \left(U\otimes V\right)^{\alpha }{}_{\beta }{}^{\gamma }=U^{\alpha }{}_{\beta }V^{\gamma }}
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19955:
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20125:
Hazewinkel, Michiel; Gubareni, Nadezhda Mikhaĭlovna; Gubareni, Nadiya; Kirichenko, Vladimir V. (2004).
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Tensor products are used in many application areas, including physics and engineering. For example, in
20319:
15497:)-bimodule, where the left and right actions are defined in the same way as the previous two examples.
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A construction of the tensor product that is basis independent can be obtained in the following way.
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is defined from their decomposition on the bases. More precisely, taking the basis decompositions of
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5810:{\displaystyle (x,y)=\left(\left(x_{1},\ldots ,x_{m}\right),\left(y_{1},\ldots ,y_{n}\right)\right)}
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20001:
may be functions instead of constants. This product of two functions is a derived function, and if
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560:
18933:{\displaystyle (f\otimes g)(x_{1},\dots ,x_{k+m})=f(x_{1},\dots ,x_{k})g(x_{k+1},\dots ,x_{k+m}).}
16419:
is not usually injective. For example, tensoring the (injective) map given by multiplication with
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7170:
This allows omitting parentheses in the tensor product of more than two vector spaces or vectors.
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1559:{\displaystyle x=\sum _{v\in B_{V}}x_{v}\,v\quad {\text{and}}\quad y=\sum _{w\in B_{W}}y_{w}\,w,}
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15641:
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8316:
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7878:
7624:{\displaystyle x_{1}\otimes \cdots \otimes x_{n}\mapsto x_{s(1)}\otimes \cdots \otimes x_{s(n)}}
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tensors, as well as tensors of mixed variance. Although in many cases such as when there is an
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9861:{\displaystyle \operatorname {Tr} A\otimes B=\operatorname {Tr} A\times \operatorname {Tr} B}
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18950:). Thus the components of the tensor product of multilinear forms can be computed by the
18206:
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is interpreted as the same polynomial, but with its coefficients regarded as elements of
16300:
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8265:
8211:{\displaystyle f\otimes g=(f\otimes Z)\circ (U\otimes g)=(V\otimes g)\circ (f\otimes W).}
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2906:, as done above. It is then straightforward to verify that with this definition, the map
866:
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is defined in exactly the same way as the tensor product of vector spaces over a field:
13296:{\displaystyle \mathrm {Hom} (U\otimes V,W)\cong \mathrm {Hom} (U,\mathrm {Hom} (V,W)).}
7694:
Tensor product of modules § Tensor product of linear maps and a change of base ring
7692:"Tensor product of linear maps" redirects here. For the generalization for modules, see
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that is spanned by the relations that the tensor product must satisfy. More precisely,
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10487:{\displaystyle (v_{1}\otimes f_{1})\otimes (v'_{1})=v_{1}\otimes v'_{1}\otimes f_{1}.}
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is the special case of the tensor product between two vectors of the same dimension.
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5534:{\displaystyle \left\{T\left(x_{i},y_{j}\right):1\leq i\leq m,1\leq j\leq n\right\}}
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5911:{\displaystyle \left(x_{i}y_{j}\right)_{\stackrel {i=1,\ldots ,m}{j=1,\ldots ,n}}}
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It is straightforward to prove that the result of this construction satisfies the
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20359:
13304:
11184:{\displaystyle (V\otimes U)^{\mu \nu }{}_{\sigma }=V^{\mu }U^{\nu }{}_{\sigma }.}
8222:
3392:
525:
16031:{\displaystyle M\otimes _{R}N=\operatorname {coker} \left(N^{J}\to N^{I}\right)}
4001:
21270:
21235:
21132:
20955:
20945:
20867:
20839:
20824:
20809:
20725:
20062:
19587:
19048:
11198:
9874:
8257:
7480:
6683:{\displaystyle Z:=\operatorname {span} \left\{f\otimes g:f\in X,g\in Y\right\}}
2338:
21215:
18537:
generalize finite-dimensional vector spaces to arbitrary dimensions. There is
5684:{\displaystyle T:\mathbb {C} ^{m}\times \mathbb {C} ^{n}\to \mathbb {C} ^{mn}}
21290:
21207:
21112:
21024:
20897:
20615:
20297:
Abo, H.; Seigal, A.; Sturmfels, B. (2015). "Eigenconfigurations of
Tensors".
20174:
18593:
Vector spaces endowed with an additional multiplicative structure are called
18534:
16816:
13932:
13576:
10306:{\displaystyle T_{s}^{r}(V)\otimes _{K}T_{s'}^{r'}(V)\to T_{s+s'}^{r+r'}(V).}
8363:
above are all two-dimensional and bases have been fixed for all of them, and
8253:
6915:
is finite-dimensional, and its dimension is the product of the dimensions of
3048:
1756:{\displaystyle B(x,y)=\sum _{v\in B_{V}}\sum _{w\in B_{W}}x_{v}y_{w}\,B(v,w)}
1200:
506:
21275:
21079:
21064:
21029:
20877:
20862:
19068:
17312:
17130:
13681:(referred to as "the canonical middle linear map"); that is, it satisfies:
7468:
4073:
3277:. To get such a vector space, one can define it as the vector space of the
3187:
3132:
581:
517:
but with tensors instead of vectors), with one tensor at each point of the
514:
510:
83:
64:
17490:. The fixed points of nonlinear maps are the eigenvectors of tensors. Let
21163:
21137:
21059:
20748:
20687:
20463:
16478:
13451:
11777:
9807:
9803:
7522:
7219:
6977:
5306:-linearly disjoint if and only if for all linearly independent sequences
4062:
3989:
considered below. (A very similar construction can be used to define the
3404:
545:
318:
31:
16256:). Colloquially, this may be rephrased by saying that a presentation of
15801:{\displaystyle M\otimes _{\mathbf {Z} }\mathbf {Z} /n\mathbf {Z} =M/nM.}
14116:
9798:
The resultant rank is at most 4, and thus the resultant dimension is 4.
6183:
with addition and scalar multiplication defined pointwise (meaning that
2330:{\displaystyle B=\sum _{v\in B_{V}}\sum _{w\in B_{W}}B(v,w)(v\otimes w)}
21044:
20507:
20428:
19993:
J's treatment also allows the representation of some tensor fields, as
19197:
does not have characteristic 2, then this definition is equivalent to:
19182:{\displaystyle V\wedge V:=V\otimes V{\big /}\{v\otimes v\mid v\in V\}.}
18576:
17126:
12507:
11721:
10509:
10149:
8249:
4080:
518:
456:
16303:. It is not in general left exact, that is, given an injective map of
13479:
More generally, the tensor product can be defined even if the ring is
21019:
20970:
20518:, vol. 211 (Revised third ed.), New York: Springer-Verlag,
13582:
The universal property also carries over, slightly modified: the map
12046:
8245:
8226:
20124:
13507:-module, and instead of the last two relations above, the relation:
3014:
that any construction of the tensor product satisfies (see below).
21049:
21034:
20303:
19609:
The symmetric algebra is constructed in a similar manner, from the
15960:
521:
20077: – Tensor product constructions for topological vector spaces
12163:{\displaystyle \langle u(a),b\rangle =\langle a,u^{*}(b)\rangle .}
2778:
This definition is quite clearly derived from the coefficients of
1829:
is uniquely and totally determined by the values that it takes on
20743:
20705:
20045:
20033:
9810:
counts the number of degrees of freedom in the resulting array).
8271:
By choosing bases of all vector spaces involved, the linear maps
7077:{\displaystyle (U\otimes V)\otimes W\cong U\otimes (V\otimes W),}
18541:, also called the "tensor product," that makes Hilbert spaces a
8587:
respectively, then the tensor product of these two matrices is:
21069:
20661:
20021:
15952:{\displaystyle \sum _{j\in J}a_{ji}m_{i}=0,\qquad a_{ij}\in R,}
10589:
9935:
9886:
309:, and the tensor product of two vectors is sometimes called an
306:
17232:{\displaystyle \psi :\mathbb {P} ^{n-1}\to \mathbb {P} ^{n-1}}
11979:{\displaystyle u(a\otimes b)=u(a)\otimes b-a\otimes u^{*}(b),}
11816:
by means of the diagonal action: for simplicity let us assume
11509:, there is a canonical map in the other direction (called the
8781:
6956:
is formed by taking all tensor products of a basis element of
18571:
Graded vector space § Operations on graded vector spaces
18288:{\displaystyle \mathbf {x} =\left(x_{1},\ldots ,x_{n}\right)}
16299:. This is referred to by saying that the tensor product is a
10700:), then the components of their tensor product are given by:
6163:
is the vector space of all complex-valued functions on a set
2209:
as a (potentially infinite) formal linear combination of the
541:
20370:
7358:, the tensor product of vectors is not commutative; that is
3312:
that have a finite number of nonzero values and identifying
20093:
20091:
12948:
11604:
4829:, which by definition means that for all positive integers
20630:
18597:. The tensor product of such algebras is described by the
12822:
8264:
do not transform injections into injections, but they are
8104:{\displaystyle (f\otimes g)(u\otimes w)=f(u)\otimes g(w).}
7173:
5028:{\displaystyle \sum _{i=1}^{n}T\left(x_{i},y_{i}\right)=0}
20624:"Bibliography on the nonabelian tensor product of groups"
18564:
13310:
12038:{\displaystyle u^{*}\in \mathrm {End} \left(V^{*}\right)}
20088:
17845:{\displaystyle \mathbb {P} ^{n-1}\to \mathbb {P} ^{n-1}}
16412:{\displaystyle M_{1}\otimes _{R}N\to M_{2}\otimes _{R}N}
12610:{\displaystyle U^{*}\otimes V\cong \mathrm {Hom} (U,V),}
11204:
8313:
is vectorized, the matrix describing the tensor product
1229:
that are nonzero at only a finite number of elements of
1071:{\displaystyle \{v\otimes w\mid v\in B_{V},w\in B_{W}\}}
564:
universal property, that is that tensor products exist.
20108:
20106:
19035:
The most general setting for the tensor product is the
18957:
11482:
on which this map is to be applied must be specified.)
7865:{\displaystyle (f\otimes W)(u\otimes w)=f(u)\otimes w.}
540:
of two vector spaces is a vector space that is defined
414:
is formed by all tensor products of a basis element of
18615:
18336:
18325:
18094:
11193:
Tensors equipped with their product operation form an
9224:
9125:
9019:
8911:
8805:
8690:
8599:
8491:
8393:
6478:{\displaystyle f\otimes g\in \mathbb {C} ^{S\times T}}
20065: – Universal construction in multilinear algebra
19777:
19619:
19554:
19514:
19443:
19401:
19361:
19203:
19120:
19094:
18779:
18759:
18739:
18684:
18629:
18488:
18323:
18303:
18237:
18209:
18182:
18159:
18139:
17858:
17802:
17762:
17710:
17686:
17623:
17585:
17565:
17496:
17472:
17452:
17415:
17393:
17373:
17347:
17325:
17297:
17273:
17251:
17183:
17141:
17111:
17088:
17036:
16963:
16860:
16722:
16577:
16538:
16359:
16317:
16270:
16233:
16198:
16171:
16139:
16099:
16048:
15969:
15877:
15832:
15744:
15684:
15644:
15594:
15560:
15526:
15462:
15377:
15338:
15265:
15226:
15177:
15146:
15118:
15055:
15010:
14721:
14683:
14673:
The universal property can be stated as follows. Let
14313:
14278:
14252:
14213:
14151:
14117:
Tensor product of modules over a non-commutative ring
14095:
14061:
14026:
13992:
13970:
13942:
13687:
13639:
13588:
13513:
13421:
13356:
13205:
13116:
13090:
13044:
12970:
12831:
12789:
12756:
12692:
12670:
12623:
12558:
12447:
12395:
12352:
12315:
12294:{\displaystyle (a\otimes b)(x)=\langle x,b\rangle a.}
12242:
12178:
12098:
12063:
11992:
11905:
11860:
11824:
11785:
11741:
11694:
11672:
11624:
11519:
11491:
11461:
11441:
11358:
11309:
11261:
11222:
11111:
11018:
10989:
10955:
10915:
10706:
10666:
10626:
10545:
10518:
10387:
10360:
10327:
10182:
10123:
9950:
9909:
9818:
8593:
8381:
8319:
8293:
8123:
8037:
7981:
7944:
7910:
7881:
7807:
7751:
7707:
7642:
7541:
7489:
7447:
7409:
7366:
7336:
7303:
7275:
7231:
7222:
in the sense that there is a canonical isomorphism:
7204:
7184:
7133:
7093:
7025:
6988:
6936:
6895:
6844:
6822:
6731:
6696:
6624:
6589:
6554:
6493:
6444:
6407:
6370:
6350:
6330:
6290:
6265:
6215:
6189:
6169:
6140:
6091:
6040:
6018:
5998:
5961:
5924:
5823:
5697:
5629:
5591:
5571:
5551:
5448:
5426:
5378:
5358:
5312:
5292:
5272:
5252:
5226:
5197:
5151:
5126:
5097:
5047:
4961:
4907:
4855:
4835:
4813:
4792:
4772:
4757:{\displaystyle \operatorname {span} \;T(X\times Y)=Z}
4725:
4703:
4683:
4663:
4643:
4611:
4573:
4553:
4525:
4476:
4448:
4383:
4340:
4291:
4245:
4215:
4187:
4135:
4107:
3960:
3926:
3880:
3842:
3788:
3734:
3416:
3354:
3318:
3286:
3255:
3227:
3195:
3158:
3079:
3057:
3027:
2990:
2964:
2912:
2883:
2854:
2819:
2784:
2547:
2521:
2495:
2446:
2438:
is nonzero at an only a finite number of elements of
2424:
2392:
2347:
2241:
2215:
2195:
1989:
1945:
1905:
1879:
1837:
1791:
1771:
1646:
1626:
1599:
1572:
1459:
1432:
1405:
1385:
1365:
1325:
1279:
1237:
1209:
1179:
1147:
1118:
1086:
1014:
983:
937:
901:
875:
827:
794:
759:
724:
698:
692:
is a vector space that has as a basis the set of all
664:
629:
600:
464:
431:
394:
352:
326:
285:
247:
216:
188:
129:
91:
43:
20598:
Topological Vector Spaces, Distributions and
Kernels
20336:
Kadison, Richard V.; Ringrose, John R. (1997).
20320:"Non-existence of tensor products of Hilbert spaces"
20103:
19499:{\displaystyle v_{1}\wedge v_{2}=-v_{2}\wedge v_{1}}
18604:
18582:
18514:
15959:
the tensor product can be computed as the following
13626:{\displaystyle \varphi :A\times B\to A\otimes _{R}B}
13195:
the tensor product is linked to the vector space of
20152:, p. 244 defines the usage "tensor product of
11724:. This map does not depend on the choice of basis.
8018:{\displaystyle f\otimes g:U\otimes W\to V\otimes Z}
7788:{\displaystyle f\otimes W:U\otimes W\to V\otimes W}
2482:, and consider the subspace of such maps instead.
20296:
19928:
19761:
19570:
19532:
19498:
19427:
19387:
19345:
19181:
19106:
19024:
18932:
18765:
18745:
18725:
18670:
18500:
18470:
18309:
18287:
18221:
18195:
18165:
18145:
18123:
17844:
17788:
17745:
17692:
17672:
17609:
17571:
17551:
17478:
17458:
17436:
17399:
17379:
17353:
17331:
17303:
17279:
17257:
17231:
17167:
17117:
17094:
17061:
17014:
16911:
16783:
16708:
16557:
16411:
16343:
16289:
16246:
16217:
16184:
16151:
16125:
16083:
16030:
15951:
15863:
15800:
15696:
15656:
15613:
15578:
15544:
15481:
15419:
15357:
15307:
15245:
15196:
15158:
15130:
15105:{\displaystyle {\overline {q}}(a\otimes b)=q(a,b)}
15104:
15041:
14994:
14707:
14663:
14299:
14264:
14234:
14199:
14101:
14079:
14045:
14004:
13976:
13954:
13915:
13669:
13625:
13555:
13442:
13407:
13295:
13154:
13102:
13076:
13038:which automatically gives the important fact that
13030:
12954:
12813:
12775:
12740:
12676:
12654:
12609:
12470:
12418:
12379:
12338:
12293:
12228:
12162:
12082:
12037:
11978:
11889:
11842:
11808:
11768:
11712:
11678:
11656:
11610:
11497:
11474:
11447:
11427:
11327:
11291:
11247:
11183:
11097:
11002:
10973:
10933:
10887:
10690:
10650:
10572:
10531:
10486:
10373:
10340:
10315:It is defined by grouping all occurring "factors"
10305:
10136:
10109:
9927:
9860:
9788:
8579:
8331:
8305:
8210:
8103:
8017:
7962:
7928:
7893:
7864:
7787:
7725:
7674:
7623:
7505:
7459:
7433:
7390:
7348:
7315:
7287:
7258:
7210:
7190:
7157:
7117:
7076:
7006:
6948:
6907:
6850:
6828:
6808:
6717:
6682:
6610:
6575:
6538:
6477:
6428:
6391:
6356:
6336:
6314:
6274:
6251:
6201:
6175:
6155:
6121:
6075:
6026:
6004:
5982:
5945:
5910:
5809:
5683:
5615:
5577:
5557:
5533:
5432:
5410:
5364:
5344:
5298:
5278:
5258:
5232:
5210:
5183:
5132:
5110:
5079:
5027:
4945:
4893:
4841:
4819:
4798:
4778:
4756:
4709:
4689:
4669:
4649:
4629:
4597:
4559:
4537:
4488:
4460:
4434:
4369:
4324:
4269:
4227:
4199:
4174:{\displaystyle {\otimes }:(v,w)\mapsto v\otimes w}
4173:
4119:
3972:
3944:
3909:
3854:
3826:
3772:
3715:
3372:
3336:
3304:
3267:
3239:
3213:
3170:
3085:
3063:
3039:
3002:
2976:
2951:{\displaystyle {\otimes }:(x,y)\mapsto x\otimes y}
2950:
2896:
2867:
2840:
2805:
2768:
2533:
2507:
2472:
2430:
2404:
2376:
2329:
2227:
2201:
2179:
1975:
1931:
1891:
1863:
1821:
1777:
1755:
1632:
1612:
1585:
1558:
1445:
1418:
1391:
1371:
1349:
1309:
1263:
1221:
1191:
1160:
1131:
1098:
1070:
995:
963:
919:
887:
853:
806:
778:
743:
710:
676:
642:
613:
524:, and each belonging to the tensor product of the
482:
443:
406:
364:
338:
297:
259:
228:
200:
174:
115:
55:
18975:Vector bundle § Operations on vector bundles
18968:
17074:
12229:{\displaystyle T_{1}^{1}(V)\to \mathrm {End} (V)}
11428:{\displaystyle T_{s}^{r}(V)\to T_{s-1}^{r-1}(V).}
2668:
2624:
2614:
2570:
865:that have a finite number of nonzero values. The
551:The tensor product can also be defined through a
21288:
20547:
19958:may have this pattern built in. For example, in
19945:
15204:a module structure under some extra conditions:
13031:{\displaystyle \dim(U\otimes V)=\dim(U)\dim(V),}
9806:i.e. the number of requisite indices (while the
7687:
531:
20338:Fundamentals of the theory of operator algebras
20335:
18946:if they are seen as multilinear maps (see also
17610:{\displaystyle n\times n\times \cdots \times n}
15633:
15042:{\displaystyle {\overline {q}}:A\otimes B\to G}
13408:{\displaystyle A\otimes _{R}B:=F(A\times B)/G,}
13307:: the tensor product is "left adjoint" to Hom.
10502:is finite dimensional, then picking a basis of
4435:{\displaystyle h(v,w)={\tilde {h}}(v\otimes w)}
1620:'s are nonzero, and find by the bilinearity of
116:{\displaystyle V\times W\rightarrow V\otimes W}
19950:
18773:their tensor product is the multilinear form:
18519:
17552:{\displaystyle A=(a_{i_{1}i_{2}\cdots i_{d}})}
14200:{\displaystyle A\otimes _{R}B:=F(A\times B)/G}
12055:, that is, in terms of the obvious pairing on
7675:{\displaystyle V^{\otimes n}\to V^{\otimes n}}
6618:are vector subspaces then the vector subspace
4370:{\displaystyle h={\tilde {h}}\circ {\otimes }}
2418:, it only remains to add the requirement that
559:, below. As for every universal property, all
20646:
20573:
20265:(lecture notes), National Taiwan University,
20048: – Second order tensor in vector algebra
19751:
19650:
19335:
19234:
19078:The exterior algebra is constructed from the
16505:be a commutative ring. The tensor product of
16488:
6980:in the sense that, given three vector spaces
82:) is a vector space to which is associated a
20576:Monoidal functors, species and Hopf algebras
20071: – Operation in mathematics and physics
19173:
19149:
17673:{\displaystyle (a_{i_{1}i_{2}\cdots i_{d}})}
15420:{\displaystyle (a\otimes b)s:=a\otimes (bs)}
15308:{\displaystyle s(a\otimes b):=(sa)\otimes b}
13149:
13136:
13130:
13117:
13071:
13045:
12808:
12790:
12770:
12757:
12655:{\displaystyle f\otimes v\in U^{*}\otimes V}
12282:
12270:
12154:
12126:
12120:
12099:
11776:may be naturally viewed as a module for the
7434:{\displaystyle x\otimes y\mapsto y\otimes x}
5545:For example, it follows immediately that if
4325:{\displaystyle {\tilde {h}}:V\otimes W\to Z}
1065:
1015:
19395:in the exterior product is usually denoted
18996:
18985:
18478:and the eigenconfiguration is given by the
17062:{\displaystyle A^{\operatorname {deg} (f)}}
16854:, the tensor product can be calculated as:
12750:Its "inverse" can be defined using a basis
12488:Given two finite dimensional vector spaces
7259:{\displaystyle V\otimes W\cong W\otimes V,}
6930:This results from the fact that a basis of
6611:{\displaystyle Y\subseteq \mathbb {C} ^{T}}
6576:{\displaystyle X\subseteq \mathbb {C} ^{S}}
3867:Then, the tensor product is defined as the
20653:
20639:
20484:
20433:"How to lose your fear of tensor products"
20282:: CS1 maint: location missing publisher (
20228:
20059: – Category admitting tensor products
18575:Some vector spaces can be decomposed into
18092:
18091:
17015:{\displaystyle A\otimes _{R}A\cong A/f(x)}
16912:{\displaystyle A\otimes _{R}B\cong B/f(x)}
12346:may be first viewed as an endomorphism of
8225:, this means that the tensor product is a
8028:is the unique linear map that satisfies:
6790:
6786:
6756:
6752:
6736:
6054:
6050:
4729:
4011:is bilinear, there is a unique linear map
3017:If arranged into a rectangular array, the
20302:
19978:the tensor product is the dyadic form of
19508:. Similar constructions are possible for
17826:
17805:
17418:
17213:
17192:
13569:is non-commutative, this is no longer an
12483:
11730:
7391:{\displaystyle v\otimes w\neq w\otimes v}
6699:
6598:
6563:
6459:
6416:
6379:
6143:
6023:
6019:
5970:
5933:
5668:
5653:
5638:
5600:
4005:Universal property of tensor product: if
2749:
2662:
2608:
2163:
2109:
1976:{\displaystyle v\otimes w:V\times W\to F}
1734:
1549:
1499:
19:For generalizations of this concept, see
21010:Covariance and contravariance of vectors
20448:. Springer Science+Business Media, LLC.
20405:
20149:
19533:{\displaystyle V\otimes \dots \otimes V}
19023:of the graphs. Compare also the section
18545:. It is essentially constructed as the
17746:{\displaystyle A\in (K^{n})^{\otimes d}}
16805:are fields containing a common subfield
12617:defined by an action of the pure tensor
7178:The tensor product of two vector spaces
6718:{\displaystyle \mathbb {C} ^{S\times T}}
4946:{\displaystyle y_{1},\ldots ,y_{n}\in Y}
4894:{\displaystyle x_{1},\ldots ,x_{n}\in X}
4000:
20443:
20317:
20177:defined, the distinction is irrelevant.
19047:A number of important subspaces of the
17680:lying in an algebraically closed field
16084:{\displaystyle N^{J}=\oplus _{j\in J}N}
13670:{\displaystyle (a,b)\mapsto a\otimes b}
13460:generated by the cartesian product and
13177:Furthermore, given three vector spaces
11292:{\displaystyle v\otimes f\mapsto f(v).}
11255:defined by its action on pure tensors:
10588:of a (tensor) product of two (or more)
7174:Commutativity as vector space operation
5372:and all linearly independent sequences
3344:with the function that takes the value
2414:. To instead have it be a proper Hamel
895:a vector space. The function that maps
556:
16:Mathematical operation on vector spaces
21289:
20592:
20462:
20381:. Universitext. Springer. p. 25.
20160:", elements of the respective modules.
20112:
20097:
19030:
18565:Tensor product of graded vector spaces
16784:{\displaystyle R\otimes _{R}R\cong R.}
15638:For vector spaces, the tensor product
13311:Tensor products of modules over a ring
13077:{\displaystyle \{u_{i}\otimes v_{j}\}}
12741:{\displaystyle (f\otimes v)(u)=f(u)v.}
12528:-vector space of all linear maps from
12387:and then viewed as an endomorphism of
11890:{\displaystyle u\in \mathrm {End} (V)}
6076:{\displaystyle x\otimes y\;:=\;T(x,y)}
3115:
2189:Then we can express any bilinear form
20634:
20600:. Mineola, N.Y.: Dover Publications.
18726:{\displaystyle g(x_{1},\dots ,x_{m})}
18671:{\displaystyle f(x_{1},\dots ,x_{k})}
18317:are the solutions of the constraint:
14715:that is bilinear, in the sense that:
14080:{\displaystyle \psi =f\circ \varphi }
12823:Evaluation map and tensor contraction
11205:Evaluation map and tensor contraction
8343:of the two matrices. For example, if
7158:{\displaystyle u\otimes (v\otimes w)}
7118:{\displaystyle (u\otimes v)\otimes w}
7016:, there is a canonical isomorphism:
6539:{\displaystyle (s,t)\mapsto f(s)g(t)}
6429:{\displaystyle g\in \mathbb {C} ^{T}}
6392:{\displaystyle f\in \mathbb {C} ^{S}}
6083:denotes this bilinear map's value at
3996:
1078:is then straightforwardly a basis of
175:{\displaystyle (v,w),\ v\in V,w\in W}
20506:
20254:Chen, Jungkai Alfred (Spring 2004),
20253:
19596:. The latter notion is the basis of
19042:
18958:Tensor product of sheaves of modules
18943:
16511:-modules applies, in particular, if
14137:-module. Then the tensor product of
11301:More generally, for tensors of type
10934:{\displaystyle U_{\beta }^{\alpha }}
8287:. Then, depending on how the tensor
7798:is the unique linear map such that:
7479:. More generally and as usual (see
5616:{\displaystyle Z:=\mathbb {C} ^{mn}}
4502:
4237:, such that, for every bilinear map
20247:
20188:"The Coevaluation on Vector Spaces"
19962:the tensor product is expressed as
18616:Tensor product of multilinear forms
16532:. In this case, the tensor product
15664:is quickly computed since bases of
13155:{\displaystyle \{u_{i}\},\{v_{j}\}}
11657:{\displaystyle v_{1},\ldots ,v_{n}}
11248:{\displaystyle V\otimes V^{*}\to K}
10172:There is a product map, called the
5983:{\displaystyle Y:=\mathbb {C} ^{n}}
5946:{\displaystyle X:=\mathbb {C} ^{m}}
5411:{\displaystyle y_{1},\ldots ,y_{n}}
5345:{\displaystyle x_{1},\ldots ,x_{m}}
5184:{\displaystyle y_{1},\ldots ,y_{n}}
5080:{\displaystyle x_{1},\ldots ,x_{n}}
3145:One considers first a vector space
2813:in the expansion by bilinearity of
2377:{\displaystyle {\text{Hom}}(V,W;F)}
372:is a sum of elementary tensors. If
346:in the sense that every element of
13:
20873:Tensors in curvilinear coordinates
20427:
20411:Elements of mathematics, Algebra I
20376:
19556:
19388:{\displaystyle v_{1}\otimes v_{2}}
17437:{\displaystyle \mathbb {P} ^{n-1}}
17339:correspond to the fixed points of
15826:, that is, a number of generators
15725:-module). The tensor product with
14360:
14319:
14109:within group isomorphism. See the
13268:
13265:
13262:
13248:
13245:
13242:
13213:
13210:
13207:
12847:
12844:
12841:
12585:
12582:
12579:
12455:
12452:
12449:
12403:
12400:
12397:
12323:
12320:
12317:
12213:
12210:
12207:
12013:
12010:
12007:
11874:
11871:
11868:
11793:
11790:
11787:
10381:for an element of the dual space:
9880:
7684:, which is called a braiding map.
6252:{\displaystyle s\mapsto f(s)+g(s)}
6122:{\displaystyle (x,y)\in X\times Y}
5191:are linearly independent then all
3827:{\displaystyle w,w_{1},w_{2}\in W}
3773:{\displaystyle v,v_{1},v_{2}\in V}
3407:the elements of one of the forms:
1822:{\displaystyle (x,y)\in V\times W}
1310:{\displaystyle (x,y)\in V\times W}
14:
21318:
20344:. Vol. I. Providence, R.I.:
20169:Analogous formulas also hold for
19428:{\displaystyle v_{1}\wedge v_{2}}
18611:Tensor product of quadratic forms
18605:Tensor product of quadratic forms
18589:Tensor product of representations
18583:Tensor product of representations
18515:Other examples of tensor products
16477:, which is not injective. Higher
15864:{\displaystyle m_{i}\in M,i\in I}
15676:immediately determine a basis of
13556:{\displaystyle (ar,b)\sim (a,rb)}
12471:{\displaystyle \mathrm {End} (V)}
12419:{\displaystyle \mathrm {End} (V)}
12339:{\displaystyle \mathrm {End} (V)}
12172:There is a canonical isomorphism
11809:{\displaystyle \mathrm {End} (V)}
10592:can be computed. For example, if
7634:induces a linear automorphism of
6725:together with the bilinear map:
6134:As another example, suppose that
4567:be complex vector spaces and let
3182:. That is, the basis elements of
2473:{\displaystyle B_{V}\times B_{W}}
1932:{\displaystyle B_{V}\times B_{W}}
1864:{\displaystyle B_{V}\times B_{W}}
1264:{\displaystyle B_{V}\times B_{W}}
964:{\displaystyle B_{V}\times B_{W}}
854:{\displaystyle B_{V}\times B_{W}}
20574:Aguiar, M.; Mahajan, S. (2010).
20468:Finite dimensional vector spaces
20439:from the original on 7 May 2021.
19435:and satisfies, by construction,
18530:Tensor product of Hilbert spaces
18447:
18419:
18396:
18239:
16262:gives rise to a presentation of
15774:
15761:
15754:
14708:{\displaystyle q:A\times B\to G}
14145:is an abelian group defined by:
10584:). In terms of these bases, the
10580:(this basis is described in the
8260:is mapped to an exact sequence (
6971:
6156:{\displaystyle \mathbb {C} ^{S}}
4598:{\displaystyle T:X\times Y\to Z}
4270:{\displaystyle h:V\times W\to Z}
1765:Hence, we see that the value of
1350:{\displaystyle B:V\times W\to F}
20578:. CRM Monograph Series Vol 29.
20342:Graduate Studies in Mathematics
20329:
20318:Garrett, Paul (July 22, 2010).
20311:
20290:
20272:from the original on 2016-03-04
20198:from the original on 2017-02-02
15926:
14677:be an abelian group with a map
14089:, and this property determines
8479:
6485:denote the function defined by
4129:, together with a bilinear map
3910:{\displaystyle V\otimes W=L/R,}
2337:making these maps similar to a
1873:. This lets us extend the maps
1509:
1503:
483:{\displaystyle V\otimes W\to Z}
25:Tensor product (disambiguation)
20229:Hungerford, Thomas W. (1974).
20222:
20209:
20192:The Unapologetic Mathematician
20180:
20163:
20143:
20118:
19923:
19835:
19733:
19707:
19317:
19291:
19011:in the category of graphs and
18969:Tensor product of line bundles
18942:This is a special case of the
18924:
18880:
18874:
18842:
18833:
18795:
18792:
18780:
18720:
18688:
18665:
18633:
18451:
18443:
18423:
18415:
18400:
18392:
17901:
17869:
17821:
17789:{\displaystyle K^{n}\to K^{n}}
17773:
17731:
17717:
17667:
17624:
17579:-dimensional tensor of format
17546:
17503:
17208:
17168:{\displaystyle K^{n}\to K^{n}}
17152:
17075:Eigenconfigurations of tensors
17054:
17048:
17009:
17003:
16992:
16986:
16906:
16900:
16889:
16883:
16775:
16763:
16754:
16748:
16732:
16726:
16700:
16674:
16668:
16642:
16636:
16610:
16604:
16578:
16558:{\displaystyle A\otimes _{R}B}
16383:
16344:{\displaystyle M_{1}\to M_{2}}
16328:
16290:{\displaystyle M\otimes _{R}N}
16133:is determined by sending some
16126:{\displaystyle N^{J}\to N^{I}}
16110:
16010:
15614:{\displaystyle A\otimes _{R}B}
15482:{\displaystyle A\otimes _{R}B}
15414:
15405:
15390:
15378:
15358:{\displaystyle A\otimes _{R}B}
15296:
15287:
15281:
15269:
15246:{\displaystyle A\otimes _{R}B}
15197:{\displaystyle A\otimes _{R}B}
15099:
15087:
15078:
15066:
15033:
14982:
14967:
14954:
14939:
14926:
14907:
14898:
14879:
14866:
14834:
14821:
14802:
14793:
14774:
14761:
14729:
14699:
14651:
14636:
14630:
14615:
14602:
14570:
14564:
14545:
14539:
14520:
14507:
14475:
14469:
14450:
14444:
14425:
14294:
14282:
14229:
14217:
14186:
14174:
14046:{\displaystyle A\otimes _{R}B}
14014:, a unique group homomorphism
13925:The first two properties make
13906:
13891:
13878:
13863:
13853:
13836:
13827:
13815:
13802:
13779:
13769:
13752:
13743:
13731:
13718:
13695:
13655:
13652:
13640:
13604:
13550:
13535:
13529:
13514:
13437:
13425:
13391:
13379:
13287:
13284:
13272:
13252:
13235:
13217:
13022:
13016:
13007:
13001:
12989:
12977:
12939:
12926:
12892:
12866:
12863:
12851:
12729:
12723:
12714:
12708:
12705:
12693:
12601:
12589:
12465:
12459:
12413:
12407:
12374:
12368:
12333:
12327:
12303:Under this isomorphism, every
12264:
12258:
12255:
12243:
12223:
12217:
12203:
12200:
12194:
12151:
12145:
12111:
12105:
12083:{\displaystyle V\otimes V^{*}}
11970:
11964:
11936:
11930:
11921:
11909:
11884:
11878:
11803:
11797:
11763:
11757:
11557:
11531:
11419:
11413:
11383:
11380:
11374:
11322:
11310:
11283:
11277:
11271:
11239:
11125:
11112:
10968:
10956:
10720:
10707:
10691:{\displaystyle G\in T_{n}^{0}}
10651:{\displaystyle F\in T_{m}^{0}}
10567:
10561:
10436:
10420:
10414:
10388:
10297:
10291:
10251:
10248:
10242:
10204:
10198:
9972:
9966:
9922:
9910:
8202:
8190:
8184:
8172:
8166:
8154:
8148:
8136:
8095:
8089:
8080:
8074:
8065:
8053:
8050:
8038:
8003:
7954:
7920:
7850:
7844:
7835:
7823:
7820:
7808:
7773:
7717:
7656:
7616:
7610:
7588:
7582:
7571:
7419:
7152:
7140:
7106:
7094:
7068:
7056:
7038:
7026:
6787:
6780:
6768:
6753:
6533:
6527:
6521:
6515:
6509:
6506:
6494:
6315:{\displaystyle s\mapsto cf(s)}
6309:
6303:
6294:
6246:
6240:
6231:
6225:
6219:
6104:
6092:
6070:
6058:
5710:
5698:
5663:
4745:
4733:
4624:
4612:
4589:
4429:
4417:
4411:
4399:
4387:
4353:
4316:
4298:
4261:
4159:
4156:
4144:
3939:
3927:
3703:
3691:
3678:
3663:
3653:
3641:
3628:
3613:
3603:
3584:
3578:
3559:
3549:
3517:
3507:
3488:
3482:
3463:
3453:
3421:
3367:
3355:
3331:
3319:
3305:{\displaystyle V\times W\to F}
3296:
3208:
3196:
3051:of the coordinate vectors of
2936:
2933:
2921:
2835:
2823:
2800:
2788:
2371:
2353:
2324:
2312:
2309:
2297:
2147:
2125:
2122:
2110:
2017:
2005:
2002:
1990:
1967:
1804:
1792:
1750:
1738:
1662:
1650:
1566:where only a finite number of
1341:
1292:
1280:
914:
902:
474:
142:
130:
101:
1:
20926:Exterior covariant derivative
20858:Tensor (intrinsic definition)
20516:Graduate Texts in Mathematics
20399:
20346:American Mathematical Society
19946:Tensor product in programming
19571:{\displaystyle \Lambda ^{n}V}
19191:When the underlying field of
19025:Tensor product of linear maps
19015:. However it is actually the
16793:A particular example is when
15719:is not a free abelian group (
12814:{\displaystyle \{u_{i}^{*}\}}
10582:article on Kronecker products
10539:naturally induces a basis of
7688:Tensor product of linear maps
7513:denote the tensor product of
7506:{\displaystyle V^{\otimes n}}
7328:On the other hand, even when
6885:are vectors spaces of finite
6863:
4099:is a vector space denoted as
2487:tensor product of two vectors
1939:as before into bilinear maps
567:
532:Definitions and constructions
20951:Raising and lowering indices
20379:An Introduction to Manifolds
19073:universal enveloping algebra
16571:-algebra itself by putting:
15634:Computing the tensor product
15504:is a commutative ring, then
15061:
15016:
14300:{\displaystyle F(A\times B)}
14235:{\displaystyle F(A\times B)}
13964:. For any middle linear map
13443:{\displaystyle F(A\times B)}
12380:{\displaystyle T_{1}^{1}(V)}
11769:{\displaystyle T_{s}^{r}(V)}
10573:{\displaystyle T_{s}^{r}(V)}
8233:of vector spaces to itself.
7532:positive integers, the map:
6868:
6027:{\displaystyle \,\otimes \,}
4677:if and only if the image of
3952:in this quotient is denoted
2485:In either construction, the
7:
21189:Gluon field strength tensor
20660:
20444:Grillet, Pierre A. (2007).
20127:Algebras, rings and modules
20039:
19956:Array programming languages
19951:Array programming languages
18948:tensors as multilinear maps
18543:symmetric monoidal category
18520:Topological tensor products
15004:Then there is a unique map
12552:. There is an isomorphism:
11003:{\displaystyle V^{\gamma }}
10174:(tensor) product of tensors
8375:are given by the matrices:
7517:copies of the vector space
7467:to itself induces a linear
5585:are positive integers then
4083:in each of its arguments):
1173:We can equivalently define
455:factors uniquely through a
10:
21323:
21000:Cartan formalism (physics)
20820:Penrose graphical notation
20075:Topological tensor product
20024:), and/or may not support
19009:category-theoretic product
19000:
18989:
18978:
18972:
18961:
18608:
18599:Littlewood–Richardson rule
18586:
18568:
18559:topological tensor product
18526:Topological tensor product
18523:
16495:Tensor product of algebras
16492:
16489:Tensor product of algebras
15697:{\displaystyle V\otimes W}
15657:{\displaystyle V\otimes W}
13321:The tensor product of two
13314:
13103:{\displaystyle U\otimes V}
9891:For non-negative integers
9884:
8332:{\displaystyle f\otimes g}
8306:{\displaystyle v\otimes w}
8262:tensor products of modules
7894:{\displaystyle W\otimes f}
7691:
7460:{\displaystyle V\otimes V}
7316:{\displaystyle w\otimes v}
7288:{\displaystyle v\otimes w}
6949:{\displaystyle V\otimes W}
6908:{\displaystyle V\otimes W}
5541:are linearly independent.
4228:{\displaystyle V\otimes W}
4120:{\displaystyle V\otimes W}
3973:{\displaystyle v\otimes w}
3040:{\displaystyle x\otimes y}
3003:{\displaystyle V\otimes W}
2228:{\displaystyle v\otimes w}
1892:{\displaystyle v\otimes w}
1192:{\displaystyle V\otimes W}
1099:{\displaystyle V\otimes W}
996:{\displaystyle v\otimes w}
931:and the other elements of
888:{\displaystyle V\otimes W}
807:{\displaystyle V\otimes W}
779:{\displaystyle w\in B_{W}}
744:{\displaystyle v\in B_{V}}
711:{\displaystyle v\otimes w}
677:{\displaystyle V\otimes W}
528:at the point with itself.
451:into another vector space
407:{\displaystyle V\otimes W}
365:{\displaystyle V\otimes W}
339:{\displaystyle V\otimes W}
298:{\displaystyle V\otimes W}
260:{\displaystyle v\otimes w}
229:{\displaystyle v\otimes w}
201:{\displaystyle V\otimes W}
56:{\displaystyle V\otimes W}
18:
21206:
21146:
21095:
21088:
20980:
20911:
20848:
20792:
20739:
20686:
20679:
20672:Glossary of tensor theory
20668:
20129:. Springer. p. 100.
19546:factors), giving rise to
19107:{\displaystyle V\wedge V}
18552:Hilbert–Schmidt operators
18501:{\displaystyle 2\times 2}
18196:{\displaystyle \psi _{i}}
15871:together with relations:
15588:. By 3), we can conclude
15171:Furthermore, we can give
14272:and G is the subgroup of
14265:{\displaystyle A\times B}
14005:{\displaystyle A\times B}
13955:{\displaystyle A\times B}
13317:Tensor product of modules
13199:linear maps, as follows:
12776:{\displaystyle \{u_{i}\}}
11713:{\displaystyle v_{i}^{*}}
11348:, there is a map, called
7972:, their tensor product:
6816:form a tensor product of
5918:form a tensor product of
4200:{\displaystyle V\times W}
3991:tensor product of modules
3171:{\displaystyle V\times W}
3106:tensor product of modules
2977:{\displaystyle V\times W}
2405:{\displaystyle V\times W}
2384:of all bilinear forms on
1222:{\displaystyle V\times W}
557:§ Universal property
505:is described through the
444:{\displaystyle V\times W}
317:. The elementary tensors
21:Tensor product of modules
21302:Operations on structures
21256:Gregorio Ricci-Curbastro
21128:Riemann curvature tensor
20835:Van der Waerden notation
20081:
19017:Kronecker tensor product
19003:Tensor product of graphs
18997:Tensor product of graphs
18992:Tensor product of fields
18986:Tensor product of fields
16813:tensor product of fields
15810:More generally, given a
14307:generated by relations:
14102:{\displaystyle \varphi }
8256:; this means that every
7963:{\displaystyle g:W\to Z}
7929:{\displaystyle f:U\to V}
7726:{\displaystyle f:U\to V}
6364:be any sets and for any
4605:be a bilinear map. Then
21226:Elwin Bruno Christoffel
21159:Angular momentum tensor
20830:Tetrad (index notation)
20800:Abstract index notation
20217:Compact closed category
19088:, the exterior product
19082:. Given a vector space
18547:metric space completion
17177:, and thus linear maps
16218:{\displaystyle a_{ij}n}
10148:(which consists of all
6962:and a basis element of
4637:is a tensor product of
4026:that makes the diagram
2958:is a bilinear map from
418:and a basis element of
241:An element of the form
21040:Levi-Civita connection
20026:higher-order functions
19930:
19763:
19572:
19534:
19500:
19429:
19389:
19347:
19183:
19108:
19051:can be constructed as
18934:
18767:
18747:
18727:
18672:
18539:an analogous operation
18502:
18472:
18311:
18297:. The eigenvectors of
18289:
18223:
18197:
18175:homogeneous polynomial
18167:
18147:
18125:
17993:
17962:
17934:
17846:
17790:
17747:
17694:
17674:
17611:
17573:
17553:
17480:
17460:
17438:
17401:
17381:
17355:
17333:
17305:
17281:
17259:
17233:
17169:
17119:
17096:
17063:
17016:
16931:. In the larger field
16913:
16846:irreducible polynomial
16815:is closely related to
16785:
16710:
16559:
16483:derived tensor product
16413:
16353:, the tensor product:
16345:
16291:
16248:
16219:
16186:
16153:
16152:{\displaystyle n\in N}
16127:
16085:
16032:
15953:
15865:
15802:
15698:
15658:
15615:
15580:
15579:{\displaystyle br:=rb}
15546:
15545:{\displaystyle ra:=ar}
15483:
15421:
15359:
15309:
15247:
15198:
15160:
15159:{\displaystyle b\in B}
15132:
15131:{\displaystyle a\in A}
15106:
15043:
14996:
14709:
14665:
14301:
14266:
14236:
14201:
14103:
14081:
14047:
14006:
13978:
13956:
13931:a bilinear map of the
13917:
13671:
13627:
13557:
13444:
13409:
13303:This is an example of
13297:
13156:
13104:
13078:
13032:
12956:
12815:
12777:
12742:
12678:
12656:
12611:
12484:Linear maps as tensors
12472:
12430:adjoint representation
12420:
12381:
12340:
12295:
12230:
12164:
12084:
12039:
11980:
11891:
11844:
11810:
11770:
11731:Adjoint representation
11714:
11680:
11658:
11612:
11499:
11485:On the other hand, if
11476:
11449:
11429:
11329:
11293:
11249:
11185:
11099:
11004:
10975:
10935:
10889:
10692:
10652:
10574:
10533:
10508:and the corresponding
10488:
10375:
10342:
10307:
10138:
10111:
9929:
9862:
9790:
8581:
8333:
8307:
8283:can be represented by
8212:
8105:
8019:
7964:
7930:
7904:Given two linear maps
7901:is defined similarly.
7895:
7866:
7789:
7727:
7676:
7625:
7507:
7461:
7435:
7392:
7350:
7317:
7289:
7260:
7212:
7192:
7159:
7119:
7078:
7008:
6976:The tensor product is
6950:
6909:
6852:
6830:
6810:
6719:
6684:
6612:
6577:
6540:
6479:
6430:
6393:
6358:
6338:
6316:
6276:
6253:
6203:
6177:
6157:
6123:
6077:
6028:
6006:
5984:
5947:
5912:
5811:
5685:
5617:
5579:
5559:
5535:
5434:
5412:
5366:
5346:
5300:
5280:
5260:
5234:
5212:
5185:
5134:
5112:
5081:
5029:
4982:
4947:
4895:
4843:
4821:
4800:
4780:
4758:
4711:
4691:
4671:
4651:
4631:
4599:
4561:
4539:
4490:
4489:{\displaystyle w\in W}
4462:
4461:{\displaystyle v\in V}
4436:
4371:
4326:
4271:
4229:
4201:
4175:
4121:
4076:is a function that is
4054:
3974:
3946:
3911:
3856:
3855:{\displaystyle s\in F}
3828:
3774:
3717:
3374:
3338:
3306:
3269:
3268:{\displaystyle w\in W}
3241:
3240:{\displaystyle v\in V}
3215:
3172:
3087:
3065:
3041:
3004:
2978:
2952:
2898:
2869:
2842:
2841:{\displaystyle B(x,y)}
2807:
2806:{\displaystyle B(v,w)}
2770:
2535:
2534:{\displaystyle y\in W}
2509:
2508:{\displaystyle x\in V}
2474:
2432:
2406:
2378:
2331:
2229:
2203:
2181:
1977:
1933:
1893:
1865:
1823:
1779:
1757:
1634:
1614:
1587:
1560:
1447:
1420:
1393:
1373:
1351:
1311:
1265:
1223:
1193:
1162:
1133:
1108:, which is called the
1100:
1072:
997:
965:
921:
889:
855:
808:
780:
745:
712:
678:
644:
615:
484:
445:
408:
366:
340:
299:
261:
230:
202:
176:
117:
57:
21297:Operations on vectors
21266:Jan Arnoldus Schouten
21221:Augustin-Louis Cauchy
20701:Differential geometry
20486:Hungerford, Thomas W.
19931:
19764:
19573:
19535:
19501:
19430:
19390:
19348:
19184:
19109:
18981:Tensor product bundle
18935:
18768:
18748:
18728:
18673:
18503:
18473:
18312:
18290:
18224:
18198:
18168:
18166:{\displaystyle \psi }
18148:
18126:
17966:
17935:
17907:
17847:
17791:
17748:
17695:
17675:
17612:
17574:
17554:
17481:
17461:
17439:
17402:
17382:
17356:
17354:{\displaystyle \psi }
17334:
17306:
17304:{\displaystyle \psi }
17282:
17260:
17234:
17170:
17120:
17097:
17064:
17022:is isomorphic (as an
17017:
16914:
16848:with coefficients in
16786:
16711:
16560:
16414:
16346:
16292:
16249:
16247:{\displaystyle N^{I}}
16220:
16187:
16185:{\displaystyle N^{J}}
16154:
16128:
16086:
16033:
15954:
15866:
15803:
15699:
15659:
15616:
15581:
15547:
15484:
15422:
15360:
15310:
15248:
15199:
15161:
15133:
15107:
15044:
14997:
14710:
14666:
14302:
14267:
14237:
14202:
14104:
14082:
14048:
14007:
13979:
13977:{\displaystyle \psi }
13957:
13918:
13672:
13628:
13575:-module, but just an
13558:
13472:-module generated by
13445:
13410:
13298:
13157:
13105:
13079:
13033:
12964:This result implies:
12957:
12816:
12778:
12743:
12679:
12657:
12612:
12473:
12421:
12382:
12341:
12296:
12231:
12165:
12085:
12040:
11981:
11892:
11845:
11843:{\displaystyle r=s=1}
11811:
11771:
11715:
11681:
11659:
11613:
11500:
11477:
11475:{\displaystyle V^{*}}
11450:
11430:
11330:
11328:{\displaystyle (r,s)}
11294:
11250:
11213:there is a canonical
11186:
11100:
11005:
10976:
10974:{\displaystyle (1,0)}
10936:
10890:
10693:
10653:
10575:
10534:
10532:{\displaystyle V^{*}}
10489:
10376:
10374:{\displaystyle f_{i}}
10343:
10341:{\displaystyle v_{i}}
10308:
10139:
10137:{\displaystyle V^{*}}
10112:
9930:
9928:{\displaystyle (r,s)}
9863:
9791:
8582:
8334:
8308:
8213:
8106:
8020:
7965:
7931:
7896:
7867:
7790:
7735:, and a vector space
7728:
7677:
7626:
7508:
7462:
7436:
7393:
7351:
7318:
7290:
7261:
7213:
7193:
7160:
7120:
7079:
7009:
7007:{\displaystyle U,V,W}
6951:
6910:
6853:
6831:
6811:
6720:
6685:
6613:
6578:
6541:
6480:
6431:
6394:
6359:
6339:
6317:
6277:
6254:
6204:
6178:
6158:
6124:
6078:
6029:
6007:
5985:
5948:
5913:
5812:
5686:
5623:and the bilinear map
5618:
5580:
5560:
5536:
5435:
5413:
5367:
5347:
5301:
5281:
5261:
5235:
5213:
5211:{\displaystyle x_{i}}
5186:
5135:
5113:
5111:{\displaystyle y_{i}}
5082:
5030:
4962:
4948:
4896:
4844:
4822:
4801:
4781:
4759:
4712:
4692:
4672:
4652:
4632:
4630:{\displaystyle (Z,T)}
4600:
4562:
4540:
4491:
4463:
4437:
4372:
4327:
4272:
4230:
4202:
4176:
4122:
4091:of two vector spaces
4057:In this section, the
4004:
3975:
3947:
3945:{\displaystyle (v,w)}
3912:
3857:
3829:
3775:
3718:
3375:
3373:{\displaystyle (v,w)}
3339:
3337:{\displaystyle (v,w)}
3307:
3270:
3242:
3216:
3214:{\displaystyle (v,w)}
3173:
3101:canonical isomorphism
3088:
3066:
3042:
3005:
2979:
2953:
2899:
2897:{\displaystyle B_{W}}
2870:
2868:{\displaystyle B_{V}}
2843:
2808:
2771:
2536:
2510:
2475:
2433:
2407:
2379:
2341:for the vector space
2332:
2230:
2204:
2182:
1978:
1934:
1894:
1866:
1824:
1780:
1758:
1635:
1615:
1613:{\displaystyle y_{w}}
1588:
1586:{\displaystyle x_{v}}
1561:
1448:
1446:{\displaystyle B_{W}}
1421:
1419:{\displaystyle B_{V}}
1394:
1374:
1352:
1312:
1273:. To see this, given
1266:
1224:
1194:
1163:
1161:{\displaystyle B_{W}}
1134:
1132:{\displaystyle B_{V}}
1101:
1073:
998:
966:
922:
920:{\displaystyle (v,w)}
890:
856:
809:
781:
746:
713:
679:
645:
643:{\displaystyle B_{W}}
616:
614:{\displaystyle B_{V}}
485:
446:
409:
367:
341:
300:
262:
231:
203:
177:
118:
58:
21241:Carl Friedrich Gauss
21174:stress–energy tensor
21169:Cauchy stress tensor
20921:Covariant derivative
20883:Antisymmetric tensor
20815:Multi-index notation
20052:Extension of scalars
19775:
19617:
19552:
19512:
19441:
19399:
19359:
19201:
19118:
19092:
19055:: these include the
18777:
18757:
18737:
18682:
18627:
18486:
18321:
18301:
18235:
18207:
18180:
18157:
18137:
17856:
17800:
17760:
17708:
17704:zero. Such a tensor
17684:
17621:
17583:
17563:
17494:
17488:algebraically closed
17470:
17450:
17413:
17391:
17371:
17345:
17323:
17315:everywhere, and the
17295:
17271:
17249:
17181:
17139:
17109:
17086:
17034:
16961:
16858:
16720:
16575:
16536:
16452:yields the zero map
16357:
16315:
16268:
16231:
16196:
16169:
16137:
16097:
16046:
15967:
15875:
15830:
15742:
15682:
15642:
15592:
15558:
15524:
15460:
15375:
15336:
15263:
15224:
15175:
15144:
15116:
15053:
15008:
14719:
14681:
14311:
14276:
14250:
14211:
14149:
14093:
14059:
14024:
13990:
13968:
13940:
13685:
13637:
13586:
13511:
13419:
13354:
13203:
13114:
13088:
13042:
12968:
12829:
12787:
12754:
12690:
12668:
12621:
12556:
12500:over the same field
12445:
12428:. In fact it is the
12393:
12350:
12313:
12240:
12176:
12096:
12061:
11990:
11903:
11858:
11822:
11783:
11739:
11692:
11670:
11622:
11517:
11489:
11459:
11439:
11356:
11307:
11259:
11220:
11209:For tensors of type
11109:
11016:
10987:
10953:
10949:be a tensor of type
10913:
10903:be a tensor of type
10704:
10664:
10624:
10543:
10516:
10385:
10358:
10325:
10180:
10163:to the ground field
10121:
9948:
9907:
9816:
8591:
8379:
8317:
8291:
8266:right exact functors
8121:
8035:
7979:
7942:
7908:
7879:
7805:
7749:
7705:
7640:
7539:
7487:
7445:
7407:
7364:
7334:
7301:
7273:
7229:
7202:
7182:
7131:
7091:
7023:
6986:
6934:
6893:
6842:
6820:
6729:
6694:
6622:
6587:
6552:
6491:
6442:
6405:
6368:
6348:
6328:
6288:
6263:
6213:
6187:
6167:
6138:
6089:
6038:
6016:
5996:
5959:
5922:
5821:
5695:
5627:
5589:
5569:
5549:
5446:
5424:
5376:
5356:
5310:
5290:
5270:
5250:
5224:
5195:
5149:
5124:
5095:
5089:linearly independent
5045:
4959:
4905:
4853:
4833:
4811:
4790:
4770:
4723:
4701:
4681:
4661:
4641:
4609:
4571:
4551:
4523:
4474:
4446:
4381:
4338:
4289:
4243:
4213:
4185:
4133:
4105:
3958:
3924:
3878:
3840:
3786:
3732:
3414:
3352:
3316:
3284:
3253:
3225:
3193:
3156:
3077:
3055:
3025:
2988:
2962:
2910:
2881:
2852:
2817:
2782:
2545:
2519:
2493:
2444:
2422:
2390:
2345:
2239:
2213:
2193:
1987:
1943:
1903:
1877:
1835:
1789:
1769:
1644:
1624:
1597:
1570:
1457:
1430:
1403:
1383:
1363:
1359:, we can decompose
1323:
1317:and a bilinear form
1277:
1235:
1207:
1177:
1145:
1116:
1084:
1012:
981:
935:
899:
873:
867:pointwise operations
825:
792:
757:
722:
696:
662:
627:
598:
462:
429:
392:
350:
324:
283:
245:
214:
186:
127:
89:
41:
21118:Nonmetricity tensor
20973:(2nd-order tensors)
20941:Hodge star operator
20931:Exterior derivative
20780:Transport phenomena
20765:Continuum mechanics
20721:Multilinear algebra
20413:. Springer-Verlag.
20348:. Thm. 2.6.4.
20263:Advanced Algebra II
20100:, pp. 403–404.
20030:Jacobian derivative
20017:is differentiable.
19031:Monoidal categories
19013:graph homomorphisms
18222:{\displaystyle d-1}
16301:right exact functor
14403: for all
12919:
12821:as in the section "
12807:
12783:and its dual basis
12367:
12193:
11756:
11735:The tensor product
11709:
11600:
11412:
11373:
10930:
10687:
10647:
10620:respectively (i.e.
10560:
10467:
10435:
10290:
10241:
10197:
9965:
7875:The tensor product
7699:Given a linear map
7349:{\displaystyle V=W}
6202:{\displaystyle f+g}
6012:will be denoted by
5691:defined by sending
4538:{\displaystyle X,Y}
4515: —
3116:As a quotient space
2235:maps according to:
503:gravitational field
315:decomposable tensor
21251:Tullio Levi-Civita
21194:Metric tensor (GR)
21108:Levi-Civita symbol
20961:Tensor contraction
20775:General relativity
20711:Euclidean geometry
20377:Tu, L. W. (2010).
20069:Tensor contraction
19926:
19827:
19820:
19759:
19568:
19530:
19496:
19425:
19385:
19343:
19179:
19104:
19021:adjacency matrices
18944:product of tensors
18930:
18763:
18743:
18733:on a vector space
18723:
18668:
18498:
18468:
18456:
18329:
18307:
18285:
18219:
18193:
18163:
18143:
18121:
18098:
17852:with coordinates:
17842:
17786:
17743:
17690:
17670:
17607:
17569:
17549:
17476:
17456:
17434:
17397:
17377:
17365:eigenconfiguration
17351:
17329:
17301:
17277:
17255:
17229:
17165:
17115:
17102:with entries in a
17092:
17059:
17012:
16909:
16781:
16706:
16555:
16409:
16341:
16287:
16244:
16215:
16182:
16149:
16123:
16081:
16028:
15949:
15893:
15861:
15798:
15694:
15654:
15611:
15576:
15542:
15520:)-bimodules where
15479:
15417:
15355:
15305:
15243:
15194:
15156:
15128:
15102:
15039:
14992:
14990:
14705:
14661:
14659:
14297:
14262:
14244:free abelian group
14232:
14197:
14099:
14077:
14043:
14002:
13974:
13952:
13913:
13911:
13667:
13623:
13553:
13489:has to be a right-
13440:
13405:
13293:
13152:
13100:
13074:
13028:
12952:
12947:
12905:
12904:
12811:
12793:
12773:
12738:
12674:
12652:
12607:
12468:
12416:
12377:
12353:
12336:
12291:
12226:
12179:
12160:
12080:
12035:
11976:
11887:
11840:
11806:
11766:
11742:
11710:
11695:
11676:
11654:
11608:
11603:
11586:
11569:
11507:finite-dimensional
11495:
11472:
11445:
11425:
11386:
11359:
11350:tensor contraction
11325:
11289:
11245:
11181:
11095:
11000:
10971:
10931:
10916:
10885:
10688:
10673:
10648:
10633:
10608:tensors of orders
10570:
10546:
10529:
10484:
10455:
10423:
10371:
10348:for an element of
10338:
10321:together: writing
10303:
10254:
10217:
10183:
10134:
10107:
10056:
10049:
10008:
10001:
9951:
9944:is an element of:
9938:on a vector space
9925:
9858:
9786:
9777:
9210:
9202:
9096:
8988:
8882:
8767:
8676:
8577:
8568:
8470:
8329:
8303:
8208:
8101:
8015:
7960:
7926:
7891:
7862:
7785:
7723:
7672:
7621:
7503:
7457:
7431:
7388:
7346:
7313:
7285:
7256:
7208:
7188:
7155:
7115:
7074:
7004:
6946:
6905:
6848:
6826:
6806:
6804:
6715:
6680:
6608:
6573:
6536:
6475:
6426:
6389:
6354:
6334:
6312:
6275:{\displaystyle cf}
6272:
6249:
6199:
6173:
6153:
6119:
6073:
6024:
6002:
5992:. Often, this map
5980:
5943:
5908:
5807:
5681:
5613:
5575:
5555:
5531:
5430:
5408:
5362:
5342:
5296:
5276:
5256:
5230:
5208:
5181:
5130:
5108:
5077:
5025:
4943:
4891:
4839:
4827:-linearly disjoint
4817:
4796:
4776:
4754:
4707:
4687:
4667:
4647:
4627:
4595:
4557:
4535:
4513:
4486:
4458:
4432:
4367:
4322:
4267:
4225:
4197:
4171:
4117:
4059:universal property
4055:
3997:Universal property
3987:universal property
3970:
3942:
3907:
3852:
3824:
3770:
3713:
3711:
3370:
3334:
3302:
3265:
3237:
3211:
3168:
3083:
3061:
3037:
3012:universal property
3000:
2974:
2948:
2894:
2865:
2838:
2803:
2766:
2764:
2728:
2705:
2651:
2597:
2531:
2505:
2470:
2428:
2402:
2374:
2327:
2293:
2270:
2225:
2199:
2177:
2078:
2050:
1973:
1929:
1889:
1861:
1819:
1775:
1753:
1713:
1690:
1630:
1610:
1583:
1556:
1538:
1488:
1443:
1416:
1389:
1369:
1347:
1307:
1261:
1219:
1189:
1158:
1129:
1096:
1068:
993:
961:
917:
885:
851:
814:is the set of the
804:
776:
741:
708:
674:
640:
611:
591:, with respective
553:universal property
499:general relativity
492:Universal property
480:
441:
404:
362:
336:
295:
257:
226:
198:
172:
113:
53:
21284:
21283:
21246:Hermann Grassmann
21202:
21201:
21154:Moment of inertia
21015:Differential form
20990:Affine connection
20805:Einstein notation
20788:
20787:
20716:Exterior calculus
20696:Coordinate system
20607:978-0-486-45352-1
20585:978-0-8218-4776-3
20525:978-0-387-95385-4
20407:Bourbaki, Nicolas
20388:978-1-4419-7399-3
20355:978-0-8218-0819-1
20136:978-1-4020-2690-4
20057:Monoidal category
19940:symmetric tensors
19799:
19797:
19611:symmetric product
19061:symmetric algebra
19043:Quotient algebras
19037:monoidal category
18952:Kronecker product
18766:{\displaystyle K}
18746:{\displaystyle V}
18622:multilinear forms
18328:
18310:{\displaystyle A}
18146:{\displaystyle n}
18133:Thus each of the
18097:
17693:{\displaystyle K}
17572:{\displaystyle d}
17479:{\displaystyle K}
17459:{\displaystyle A}
17400:{\displaystyle n}
17380:{\displaystyle A}
17332:{\displaystyle A}
17280:{\displaystyle A}
17258:{\displaystyle K}
17241:projective spaces
17118:{\displaystyle K}
17095:{\displaystyle A}
17028:-algebra) to the
15878:
15456:)-bimodule, then
15332:)-bimodule, then
15220:)-bimodule, then
15064:
15019:
14404:
13679:middle linear map
13084:forms a basis of
12895:
12677:{\displaystyle U}
12662:on an element of
11852:, then, for each
11679:{\displaystyle V}
11560:
11498:{\displaystyle V}
11448:{\displaystyle V}
10146:dual vector space
10014:
10012:
9980:
9978:
9802:here denotes the
8341:Kronecker product
7471:that is called a
7211:{\displaystyle W}
7191:{\displaystyle V}
6851:{\displaystyle Y}
6829:{\displaystyle X}
6357:{\displaystyle T}
6337:{\displaystyle S}
6176:{\displaystyle S}
6005:{\displaystyle T}
5904:
5578:{\displaystyle n}
5558:{\displaystyle m}
5433:{\displaystyle Y}
5365:{\displaystyle X}
5299:{\displaystyle T}
5279:{\displaystyle Y}
5259:{\displaystyle X}
5233:{\displaystyle 0}
5133:{\displaystyle 0}
4849:and all elements
4842:{\displaystyle n}
4820:{\displaystyle T}
4799:{\displaystyle Y}
4779:{\displaystyle X}
4710:{\displaystyle Z}
4690:{\displaystyle T}
4670:{\displaystyle Y}
4650:{\displaystyle X}
4560:{\displaystyle Z}
4511:
4503:Linearly disjoint
4414:
4356:
4301:
3920:and the image of
3151:Cartesian product
3086:{\displaystyle y}
3064:{\displaystyle x}
3019:coordinate vector
2706:
2683:
2629:
2575:
2431:{\displaystyle B}
2351:
2271:
2248:
2202:{\displaystyle B}
2051:
2023:
1778:{\displaystyle B}
1691:
1668:
1633:{\displaystyle B}
1516:
1507:
1466:
1392:{\displaystyle y}
1372:{\displaystyle x}
1199:to be the set of
820:Cartesian product
311:elementary tensor
182:to an element of
150:
123:that maps a pair
21314:
21261:Bernhard Riemann
21093:
21092:
20936:Exterior product
20903:Two-point tensor
20888:Symmetric tensor
20770:Electromagnetism
20684:
20683:
20655:
20648:
20641:
20632:
20631:
20627:
20619:
20594:Trèves, François
20589:
20570:
20544:
20503:
20481:
20459:
20446:Abstract Algebra
20440:
20424:
20393:
20392:
20374:
20368:
20367:
20333:
20327:
20326:
20324:
20315:
20309:
20308:
20306:
20294:
20288:
20287:
20281:
20273:
20271:
20260:
20256:"Tensor product"
20251:
20245:
20244:
20226:
20220:
20213:
20207:
20206:
20204:
20203:
20184:
20178:
20167:
20161:
20147:
20141:
20140:
20122:
20116:
20110:
20101:
20095:
20016:
20008:
20004:
20000:
19996:
19989:
19985:
19981:
19973:
19969:
19965:
19935:
19933:
19932:
19927:
19916:
19915:
19903:
19902:
19872:
19871:
19853:
19852:
19834:
19833:
19826:
19821:
19816:
19787:
19786:
19771:More generally:
19768:
19766:
19765:
19760:
19755:
19754:
19748:
19747:
19732:
19731:
19719:
19718:
19703:
19702:
19690:
19689:
19677:
19676:
19664:
19663:
19654:
19653:
19647:
19646:
19603:
19595:
19585:
19579:
19577:
19575:
19574:
19569:
19564:
19563:
19545:
19539:
19537:
19536:
19531:
19507:
19505:
19503:
19502:
19497:
19495:
19494:
19482:
19481:
19466:
19465:
19453:
19452:
19434:
19432:
19431:
19426:
19424:
19423:
19411:
19410:
19394:
19392:
19391:
19386:
19384:
19383:
19371:
19370:
19352:
19350:
19349:
19344:
19339:
19338:
19332:
19331:
19316:
19315:
19303:
19302:
19287:
19286:
19274:
19273:
19261:
19260:
19248:
19247:
19238:
19237:
19231:
19230:
19196:
19188:
19186:
19185:
19180:
19148:
19147:
19113:
19111:
19110:
19105:
19087:
19080:exterior product
19065:Clifford algebra
19057:exterior algebra
18964:Sheaf of modules
18939:
18937:
18936:
18931:
18923:
18922:
18898:
18897:
18873:
18872:
18854:
18853:
18832:
18831:
18807:
18806:
18772:
18770:
18769:
18764:
18752:
18750:
18749:
18744:
18732:
18730:
18729:
18724:
18719:
18718:
18700:
18699:
18677:
18675:
18674:
18669:
18664:
18663:
18645:
18644:
18511:of this matrix.
18507:
18505:
18504:
18499:
18477:
18475:
18474:
18469:
18461:
18460:
18450:
18442:
18441:
18422:
18414:
18413:
18399:
18391:
18390:
18377:
18376:
18360:
18359:
18348:
18347:
18330:
18326:
18316:
18314:
18313:
18308:
18296:
18294:
18292:
18291:
18286:
18284:
18280:
18279:
18278:
18260:
18259:
18242:
18228:
18226:
18225:
18220:
18202:
18200:
18199:
18194:
18192:
18191:
18172:
18170:
18169:
18164:
18152:
18150:
18149:
18144:
18130:
18128:
18127:
18122:
18099:
18095:
18090:
18089:
18088:
18087:
18070:
18069:
18068:
18067:
18053:
18052:
18051:
18050:
18036:
18035:
18034:
18033:
18021:
18020:
18011:
18010:
17992:
17987:
17980:
17979:
17961:
17956:
17949:
17948:
17933:
17928:
17921:
17920:
17900:
17899:
17881:
17880:
17868:
17867:
17851:
17849:
17848:
17843:
17841:
17840:
17829:
17820:
17819:
17808:
17795:
17793:
17792:
17787:
17785:
17784:
17772:
17771:
17752:
17750:
17749:
17744:
17742:
17741:
17729:
17728:
17699:
17697:
17696:
17691:
17679:
17677:
17676:
17671:
17666:
17665:
17664:
17663:
17651:
17650:
17641:
17640:
17616:
17614:
17613:
17608:
17578:
17576:
17575:
17570:
17558:
17556:
17555:
17550:
17545:
17544:
17543:
17542:
17530:
17529:
17520:
17519:
17485:
17483:
17482:
17477:
17465:
17463:
17462:
17457:
17445:
17443:
17441:
17440:
17435:
17433:
17432:
17421:
17406:
17404:
17403:
17398:
17386:
17384:
17383:
17378:
17362:
17360:
17358:
17357:
17352:
17338:
17336:
17335:
17330:
17310:
17308:
17307:
17302:
17286:
17284:
17283:
17278:
17266:
17264:
17262:
17261:
17256:
17238:
17236:
17235:
17230:
17228:
17227:
17216:
17207:
17206:
17195:
17176:
17174:
17172:
17171:
17166:
17164:
17163:
17151:
17150:
17124:
17122:
17121:
17116:
17101:
17099:
17098:
17093:
17070:
17068:
17066:
17065:
17060:
17058:
17057:
17027:
17021:
17019:
17018:
17013:
16999:
16976:
16975:
16956:
16949:Galois extension
16946:
16936:
16930:
16924:
16918:
16916:
16915:
16910:
16896:
16873:
16872:
16853:
16843:
16837:
16810:
16804:
16798:
16790:
16788:
16787:
16782:
16744:
16743:
16715:
16713:
16712:
16707:
16699:
16698:
16686:
16685:
16667:
16666:
16654:
16653:
16635:
16634:
16622:
16621:
16603:
16602:
16590:
16589:
16570:
16564:
16562:
16561:
16556:
16551:
16550:
16529:
16522:
16516:
16510:
16504:
16476:
16451:
16438:
16424:
16418:
16416:
16415:
16410:
16405:
16404:
16395:
16394:
16379:
16378:
16369:
16368:
16352:
16350:
16348:
16347:
16342:
16340:
16339:
16327:
16326:
16308:
16298:
16296:
16294:
16293:
16288:
16283:
16282:
16261:
16255:
16253:
16251:
16250:
16245:
16243:
16242:
16224:
16222:
16221:
16216:
16211:
16210:
16191:
16189:
16188:
16183:
16181:
16180:
16164:
16158:
16156:
16155:
16150:
16132:
16130:
16129:
16124:
16122:
16121:
16109:
16108:
16092:
16090:
16088:
16087:
16082:
16077:
16076:
16058:
16057:
16037:
16035:
16034:
16029:
16027:
16023:
16022:
16021:
16009:
16008:
15982:
15981:
15958:
15956:
15955:
15950:
15939:
15938:
15916:
15915:
15906:
15905:
15892:
15870:
15868:
15867:
15862:
15842:
15841:
15825:
15819:
15807:
15805:
15804:
15799:
15788:
15777:
15769:
15764:
15759:
15758:
15757:
15737:
15724:
15718:
15705:
15703:
15701:
15700:
15695:
15675:
15669:
15663:
15661:
15660:
15655:
15620:
15618:
15617:
15612:
15607:
15606:
15587:
15585:
15583:
15582:
15577:
15551:
15549:
15548:
15543:
15488:
15486:
15485:
15480:
15475:
15474:
15428:
15426:
15424:
15423:
15418:
15364:
15362:
15361:
15356:
15351:
15350:
15316:
15314:
15312:
15311:
15306:
15252:
15250:
15249:
15244:
15239:
15238:
15203:
15201:
15200:
15195:
15190:
15189:
15167:
15165:
15163:
15162:
15157:
15137:
15135:
15134:
15129:
15111:
15109:
15108:
15103:
15065:
15057:
15048:
15046:
15045:
15040:
15020:
15012:
15001:
14999:
14998:
14993:
14991:
14925:
14924:
14897:
14896:
14865:
14864:
14852:
14851:
14814:
14813:
14786:
14785:
14754:
14753:
14741:
14740:
14714:
14712:
14711:
14706:
14670:
14668:
14667:
14662:
14660:
14611:
14601:
14600:
14588:
14587:
14563:
14562:
14538:
14537:
14516:
14500:
14499:
14487:
14486:
14462:
14461:
14437:
14436:
14421:
14405:
14402:
14391:
14390:
14378:
14377:
14350:
14349:
14337:
14336:
14317:
14306:
14304:
14303:
14298:
14271:
14269:
14268:
14263:
14241:
14239:
14238:
14233:
14206:
14204:
14203:
14198:
14193:
14164:
14163:
14108:
14106:
14105:
14100:
14088:
14086:
14084:
14083:
14078:
14052:
14050:
14049:
14044:
14039:
14038:
14019:
14013:
14011:
14009:
14008:
14003:
13983:
13981:
13980:
13975:
13963:
13961:
13959:
13958:
13953:
13930:
13922:
13920:
13919:
13914:
13912:
13852:
13801:
13762:
13711:
13676:
13674:
13673:
13668:
13632:
13630:
13629:
13624:
13619:
13618:
13574:
13568:
13562:
13560:
13559:
13554:
13506:
13500:
13494:
13488:
13471:
13465:
13457:
13449:
13447:
13446:
13441:
13414:
13412:
13411:
13406:
13398:
13369:
13368:
13349:
13335:
13329:
13305:adjoint functors
13302:
13300:
13299:
13294:
13271:
13251:
13216:
13194:
13188:
13182:
13173:
13167:
13161:
13159:
13158:
13153:
13148:
13147:
13129:
13128:
13109:
13107:
13106:
13101:
13083:
13081:
13080:
13075:
13070:
13069:
13057:
13056:
13037:
13035:
13034:
13029:
12961:
12959:
12958:
12953:
12951:
12950:
12938:
12937:
12918:
12913:
12903:
12878:
12877:
12850:
12820:
12818:
12817:
12812:
12806:
12801:
12782:
12780:
12779:
12774:
12769:
12768:
12747:
12745:
12744:
12739:
12685:
12683:
12681:
12680:
12675:
12661:
12659:
12658:
12653:
12645:
12644:
12616:
12614:
12613:
12608:
12588:
12568:
12567:
12551:
12539:
12533:
12527:
12521:
12515:
12505:
12499:
12493:
12479:
12477:
12475:
12474:
12469:
12458:
12438:
12427:
12425:
12423:
12422:
12417:
12406:
12386:
12384:
12383:
12378:
12366:
12361:
12345:
12343:
12342:
12337:
12326:
12308:
12300:
12298:
12297:
12292:
12235:
12233:
12232:
12227:
12216:
12192:
12187:
12169:
12167:
12166:
12161:
12144:
12143:
12091:
12089:
12087:
12086:
12081:
12079:
12078:
12054:
12044:
12042:
12041:
12036:
12034:
12030:
12029:
12016:
12002:
12001:
11985:
11983:
11982:
11977:
11963:
11962:
11898:
11896:
11894:
11893:
11888:
11877:
11851:
11849:
11847:
11846:
11841:
11815:
11813:
11812:
11807:
11796:
11775:
11773:
11772:
11767:
11755:
11750:
11719:
11717:
11716:
11711:
11708:
11703:
11687:
11685:
11683:
11682:
11677:
11664:is any basis of
11663:
11661:
11660:
11655:
11653:
11652:
11634:
11633:
11617:
11615:
11614:
11609:
11607:
11606:
11599:
11594:
11582:
11581:
11568:
11549:
11548:
11511:coevaluation map
11504:
11502:
11501:
11496:
11481:
11479:
11478:
11473:
11471:
11470:
11454:
11452:
11451:
11446:
11434:
11432:
11431:
11426:
11411:
11400:
11372:
11367:
11347:
11336:
11334:
11332:
11331:
11326:
11298:
11296:
11295:
11290:
11254:
11252:
11251:
11246:
11238:
11237:
11212:
11190:
11188:
11187:
11182:
11177:
11176:
11171:
11168:
11167:
11158:
11157:
11145:
11144:
11139:
11136:
11135:
11104:
11102:
11101:
11096:
11094:
11093:
11084:
11083:
11078:
11075:
11074:
11062:
11061:
11056:
11053:
11052:
11047:
11044:
11043:
11038:
11034:
11011:
11009:
11007:
11006:
11001:
10999:
10998:
10981:with components
10980:
10978:
10977:
10972:
10948:
10942:
10940:
10938:
10937:
10932:
10929:
10924:
10907:with components
10906:
10902:
10894:
10892:
10891:
10886:
10881:
10880:
10879:
10878:
10860:
10859:
10844:
10843:
10828:
10827:
10807:
10806:
10805:
10804:
10792:
10791:
10782:
10781:
10764:
10763:
10762:
10761:
10743:
10742:
10733:
10732:
10699:
10697:
10695:
10694:
10689:
10686:
10681:
10657:
10655:
10654:
10649:
10646:
10641:
10619:
10613:
10603:
10597:
10579:
10577:
10576:
10571:
10559:
10554:
10538:
10536:
10535:
10530:
10528:
10527:
10507:
10501:
10493:
10491:
10490:
10485:
10480:
10479:
10463:
10451:
10450:
10431:
10413:
10412:
10400:
10399:
10380:
10378:
10377:
10372:
10370:
10369:
10353:
10347:
10345:
10344:
10339:
10337:
10336:
10320:
10312:
10310:
10309:
10304:
10289:
10288:
10273:
10272:
10240:
10239:
10230:
10229:
10216:
10215:
10196:
10191:
10168:
10162:
10156:
10143:
10141:
10140:
10135:
10133:
10132:
10116:
10114:
10113:
10108:
10103:
10102:
10094:
10090:
10089:
10072:
10071:
10055:
10050:
10045:
10044:
10043:
10025:
10024:
10007:
10002:
9997:
9964:
9959:
9943:
9934:
9932:
9931:
9926:
9902:
9896:
9869:
9867:
9865:
9864:
9859:
9795:
9793:
9792:
9787:
9782:
9781:
9774:
9773:
9758:
9757:
9740:
9739:
9724:
9723:
9706:
9705:
9690:
9689:
9672:
9671:
9656:
9655:
9636:
9635:
9620:
9619:
9602:
9601:
9586:
9585:
9568:
9567:
9552:
9551:
9534:
9533:
9518:
9517:
9498:
9497:
9482:
9481:
9464:
9463:
9448:
9447:
9430:
9429:
9414:
9413:
9396:
9395:
9380:
9379:
9360:
9359:
9344:
9343:
9326:
9325:
9310:
9309:
9292:
9291:
9276:
9275:
9258:
9257:
9242:
9241:
9215:
9214:
9207:
9206:
9199:
9198:
9181:
9180:
9161:
9160:
9143:
9142:
9119:
9118:
9101:
9100:
9093:
9092:
9075:
9074:
9055:
9054:
9037:
9036:
9013:
9012:
8993:
8992:
8985:
8984:
8967:
8966:
8947:
8946:
8929:
8928:
8905:
8904:
8887:
8886:
8879:
8878:
8861:
8860:
8841:
8840:
8823:
8822:
8799:
8798:
8772:
8771:
8764:
8763:
8746:
8745:
8726:
8725:
8708:
8707:
8681:
8680:
8673:
8672:
8655:
8654:
8635:
8634:
8617:
8616:
8586:
8584:
8583:
8578:
8573:
8572:
8565:
8564:
8547:
8546:
8527:
8526:
8509:
8508:
8475:
8474:
8467:
8466:
8449:
8448:
8429:
8428:
8411:
8410:
8374:
8368:
8362:
8356:
8338:
8336:
8335:
8330:
8312:
8310:
8309:
8304:
8282:
8276:
8243:
8239:
8217:
8215:
8214:
8209:
8110:
8108:
8107:
8102:
8024:
8022:
8021:
8016:
7971:
7969:
7967:
7966:
7961:
7935:
7933:
7932:
7927:
7900:
7898:
7897:
7892:
7871:
7869:
7868:
7863:
7794:
7792:
7791:
7786:
7738:
7734:
7732:
7730:
7729:
7724:
7683:
7681:
7679:
7678:
7673:
7671:
7670:
7655:
7654:
7630:
7628:
7627:
7622:
7620:
7619:
7592:
7591:
7570:
7569:
7551:
7550:
7531:
7527:
7520:
7516:
7512:
7510:
7509:
7504:
7502:
7501:
7477:
7476:
7466:
7464:
7463:
7458:
7440:
7438:
7437:
7432:
7399:
7397:
7395:
7394:
7389:
7357:
7355:
7353:
7352:
7347:
7324:
7322:
7320:
7319:
7314:
7294:
7292:
7291:
7286:
7265:
7263:
7262:
7257:
7217:
7215:
7214:
7209:
7197:
7195:
7194:
7189:
7166:
7164:
7162:
7161:
7156:
7124:
7122:
7121:
7116:
7083:
7081:
7080:
7075:
7015:
7013:
7011:
7010:
7005:
6967:
6961:
6955:
6953:
6952:
6947:
6926:
6920:
6914:
6912:
6911:
6906:
6884:
6878:
6859:
6857:
6855:
6854:
6849:
6835:
6833:
6832:
6827:
6815:
6813:
6812:
6807:
6805:
6784:
6766:
6765:
6750:
6738:
6724:
6722:
6721:
6716:
6714:
6713:
6702:
6689:
6687:
6686:
6681:
6679:
6675:
6617:
6615:
6614:
6609:
6607:
6606:
6601:
6582:
6580:
6579:
6574:
6572:
6571:
6566:
6547:
6545:
6543:
6542:
6537:
6484:
6482:
6481:
6476:
6474:
6473:
6462:
6437:
6435:
6433:
6432:
6427:
6425:
6424:
6419:
6398:
6396:
6395:
6390:
6388:
6387:
6382:
6363:
6361:
6360:
6355:
6343:
6341:
6340:
6335:
6323:
6321:
6319:
6318:
6313:
6281:
6279:
6278:
6273:
6258:
6256:
6255:
6250:
6208:
6206:
6205:
6200:
6182:
6180:
6179:
6174:
6162:
6160:
6159:
6154:
6152:
6151:
6146:
6130:
6128:
6126:
6125:
6120:
6082:
6080:
6079:
6074:
6033:
6031:
6030:
6025:
6011:
6009:
6008:
6003:
5991:
5989:
5987:
5986:
5981:
5979:
5978:
5973:
5952:
5950:
5949:
5944:
5942:
5941:
5936:
5917:
5915:
5914:
5909:
5907:
5906:
5905:
5903:
5880:
5857:
5854:
5850:
5849:
5848:
5839:
5838:
5816:
5814:
5813:
5808:
5806:
5802:
5801:
5797:
5796:
5795:
5777:
5776:
5759:
5755:
5754:
5753:
5735:
5734:
5690:
5688:
5687:
5682:
5680:
5679:
5671:
5662:
5661:
5656:
5647:
5646:
5641:
5622:
5620:
5619:
5614:
5612:
5611:
5603:
5584:
5582:
5581:
5576:
5564:
5562:
5561:
5556:
5540:
5538:
5537:
5532:
5530:
5526:
5489:
5485:
5484:
5483:
5471:
5470:
5441:
5439:
5437:
5436:
5431:
5417:
5415:
5414:
5409:
5407:
5406:
5388:
5387:
5371:
5369:
5368:
5363:
5351:
5349:
5348:
5343:
5341:
5340:
5322:
5321:
5305:
5303:
5302:
5297:
5285:
5283:
5282:
5277:
5265:
5263:
5262:
5257:
5241:
5239:
5237:
5236:
5231:
5217:
5215:
5214:
5209:
5207:
5206:
5190:
5188:
5187:
5182:
5180:
5179:
5161:
5160:
5141:
5139:
5137:
5136:
5131:
5117:
5115:
5114:
5109:
5107:
5106:
5086:
5084:
5083:
5078:
5076:
5075:
5057:
5056:
5036:
5034:
5032:
5031:
5026:
5018:
5014:
5013:
5012:
5000:
4999:
4981:
4976:
4952:
4950:
4949:
4944:
4936:
4935:
4917:
4916:
4900:
4898:
4897:
4892:
4884:
4883:
4865:
4864:
4848:
4846:
4845:
4840:
4826:
4824:
4823:
4818:
4805:
4803:
4802:
4797:
4785:
4783:
4782:
4777:
4765:
4763:
4761:
4760:
4755:
4716:
4714:
4713:
4708:
4696:
4694:
4693:
4688:
4676:
4674:
4673:
4668:
4656:
4654:
4653:
4648:
4636:
4634:
4633:
4628:
4604:
4602:
4601:
4596:
4566:
4564:
4563:
4558:
4546:
4544:
4542:
4541:
4536:
4516:
4497:
4495:
4493:
4492:
4487:
4467:
4465:
4464:
4459:
4441:
4439:
4438:
4433:
4416:
4415:
4407:
4376:
4374:
4373:
4368:
4366:
4358:
4357:
4349:
4333:
4331:
4329:
4328:
4323:
4303:
4302:
4294:
4278:
4276:
4274:
4273:
4268:
4236:
4234:
4232:
4231:
4226:
4206:
4204:
4203:
4198:
4180:
4178:
4177:
4172:
4140:
4128:
4126:
4124:
4123:
4118:
4098:
4094:
4052:
4047:
4046:
4045:
4040:
4025:
4024:
4023:
4022:
4017:
4010:
3981:
3979:
3977:
3976:
3971:
3951:
3949:
3948:
3943:
3916:
3914:
3913:
3908:
3900:
3863:
3861:
3859:
3858:
3853:
3833:
3831:
3830:
3825:
3817:
3816:
3804:
3803:
3781:
3779:
3777:
3776:
3771:
3763:
3762:
3750:
3749:
3722:
3720:
3719:
3714:
3712:
3602:
3601:
3577:
3576:
3548:
3547:
3535:
3534:
3500:
3499:
3475:
3474:
3446:
3445:
3433:
3432:
3402:
3398:
3390:
3383:
3379:
3377:
3376:
3371:
3347:
3343:
3341:
3340:
3335:
3311:
3309:
3308:
3303:
3276:
3274:
3272:
3271:
3266:
3246:
3244:
3243:
3238:
3220:
3218:
3217:
3212:
3185:
3177:
3175:
3174:
3169:
3148:
3140:
3130:
3126:
3094:
3092:
3090:
3089:
3084:
3070:
3068:
3067:
3062:
3046:
3044:
3043:
3038:
3009:
3007:
3006:
3001:
2983:
2981:
2980:
2975:
2957:
2955:
2954:
2949:
2917:
2905:
2903:
2901:
2900:
2895:
2893:
2892:
2874:
2872:
2871:
2866:
2864:
2863:
2848:using the bases
2847:
2845:
2844:
2839:
2812:
2810:
2809:
2804:
2775:
2773:
2772:
2767:
2765:
2748:
2747:
2738:
2737:
2727:
2726:
2725:
2704:
2703:
2702:
2676:
2672:
2671:
2661:
2660:
2650:
2649:
2648:
2628:
2627:
2618:
2617:
2607:
2606:
2596:
2595:
2594:
2574:
2573:
2540:
2538:
2537:
2532:
2514:
2512:
2511:
2506:
2481:
2479:
2477:
2476:
2471:
2469:
2468:
2456:
2455:
2437:
2435:
2434:
2429:
2413:
2411:
2409:
2408:
2403:
2383:
2381:
2380:
2375:
2352:
2349:
2336:
2334:
2333:
2328:
2292:
2291:
2290:
2269:
2268:
2267:
2234:
2232:
2231:
2226:
2208:
2206:
2205:
2200:
2186:
2184:
2183:
2178:
2173:
2172:
2162:
2161:
2146:
2135:
2108:
2107:
2106:
2093:
2092:
2091:
2077:
2076:
2075:
2063:
2049:
2048:
2047:
2035:
1982:
1980:
1979:
1974:
1938:
1936:
1935:
1930:
1928:
1927:
1915:
1914:
1898:
1896:
1895:
1890:
1872:
1870:
1868:
1867:
1862:
1860:
1859:
1847:
1846:
1828:
1826:
1825:
1820:
1784:
1782:
1781:
1776:
1762:
1760:
1759:
1754:
1733:
1732:
1723:
1722:
1712:
1711:
1710:
1689:
1688:
1687:
1639:
1637:
1636:
1631:
1619:
1617:
1616:
1611:
1609:
1608:
1592:
1590:
1589:
1584:
1582:
1581:
1565:
1563:
1562:
1557:
1548:
1547:
1537:
1536:
1535:
1508:
1505:
1498:
1497:
1487:
1486:
1485:
1452:
1450:
1449:
1444:
1442:
1441:
1425:
1423:
1422:
1417:
1415:
1414:
1398:
1396:
1395:
1390:
1378:
1376:
1375:
1370:
1358:
1356:
1354:
1353:
1348:
1316:
1314:
1313:
1308:
1272:
1270:
1268:
1267:
1262:
1260:
1259:
1247:
1246:
1228:
1226:
1225:
1220:
1198:
1196:
1195:
1190:
1169:
1167:
1165:
1164:
1159:
1157:
1156:
1138:
1136:
1135:
1130:
1128:
1127:
1107:
1105:
1103:
1102:
1097:
1077:
1075:
1074:
1069:
1064:
1063:
1045:
1044:
1004:
1002:
1000:
999:
994:
974:
970:
968:
967:
962:
960:
959:
947:
946:
930:
926:
924:
923:
918:
894:
892:
891:
886:
864:
860:
858:
857:
852:
850:
849:
837:
836:
813:
811:
810:
805:
787:
785:
783:
782:
777:
775:
774:
750:
748:
747:
742:
740:
739:
717:
715:
714:
709:
691:
687:
683:
681:
680:
675:
651:
649:
647:
646:
641:
639:
638:
620:
618:
617:
612:
610:
609:
590:
579:
575:
489:
487:
486:
481:
454:
450:
448:
447:
442:
421:
417:
413:
411:
410:
405:
387:
381:
371:
369:
368:
363:
345:
343:
342:
337:
304:
302:
301:
296:
279:. An element of
278:
274:
266:
264:
263:
258:
237:
235:
233:
232:
227:
207:
205:
204:
199:
181:
179:
178:
173:
148:
122:
120:
119:
114:
77:
71:
62:
60:
59:
54:
21322:
21321:
21317:
21316:
21315:
21313:
21312:
21311:
21287:
21286:
21285:
21280:
21231:Albert Einstein
21198:
21179:Einstein tensor
21142:
21123:Ricci curvature
21103:Kronecker delta
21089:Notable tensors
21084:
21005:Connection form
20982:
20976:
20907:
20893:Tensor operator
20850:
20844:
20784:
20760:Computer vision
20753:
20735:
20731:Tensor calculus
20675:
20664:
20659:
20622:
20608:
20586:
20567:
20559:. AMS Chelsea.
20526:
20500:
20478:
20456:
20429:Gowers, Timothy
20421:
20402:
20397:
20396:
20389:
20375:
20371:
20356:
20334:
20330:
20322:
20316:
20312:
20295:
20291:
20275:
20274:
20269:
20258:
20252:
20248:
20241:
20227:
20223:
20214:
20210:
20201:
20199:
20186:
20185:
20181:
20168:
20164:
20150:Bourbaki (1989)
20148:
20144:
20137:
20123:
20119:
20115:, pp. 407.
20111:
20104:
20096:
20089:
20084:
20042:
20014:
20006:
20002:
19998:
19994:
19987:
19983:
19979:
19971:
19967:
19963:
19953:
19948:
19911:
19907:
19892:
19888:
19861:
19857:
19848:
19844:
19829:
19828:
19822:
19800:
19798:
19782:
19778:
19776:
19773:
19772:
19750:
19749:
19743:
19739:
19727:
19723:
19714:
19710:
19698:
19694:
19685:
19681:
19672:
19668:
19659:
19655:
19649:
19648:
19642:
19641:
19618:
19615:
19614:
19599:
19591:
19581:
19559:
19555:
19553:
19550:
19549:
19547:
19541:
19513:
19510:
19509:
19490:
19486:
19477:
19473:
19461:
19457:
19448:
19444:
19442:
19439:
19438:
19436:
19419:
19415:
19406:
19402:
19400:
19397:
19396:
19379:
19375:
19366:
19362:
19360:
19357:
19356:
19334:
19333:
19327:
19323:
19311:
19307:
19298:
19294:
19282:
19278:
19269:
19265:
19256:
19252:
19243:
19239:
19233:
19232:
19226:
19225:
19202:
19199:
19198:
19192:
19143:
19142:
19119:
19116:
19115:
19114:is defined as:
19093:
19090:
19089:
19083:
19045:
19033:
19005:
18999:
18994:
18988:
18983:
18977:
18971:
18966:
18960:
18912:
18908:
18887:
18883:
18868:
18864:
18849:
18845:
18821:
18817:
18802:
18798:
18778:
18775:
18774:
18758:
18755:
18754:
18753:over the field
18738:
18735:
18734:
18714:
18710:
18695:
18691:
18683:
18680:
18679:
18659:
18655:
18640:
18636:
18628:
18625:
18624:
18618:
18613:
18607:
18591:
18585:
18573:
18567:
18532:
18524:Main articles:
18522:
18517:
18487:
18484:
18483:
18455:
18454:
18446:
18437:
18433:
18431:
18426:
18418:
18409:
18405:
18403:
18395:
18386:
18382:
18379:
18378:
18372:
18368:
18366:
18361:
18355:
18351:
18349:
18343:
18339:
18332:
18331:
18324:
18322:
18319:
18318:
18302:
18299:
18298:
18274:
18270:
18255:
18251:
18250:
18246:
18238:
18236:
18233:
18232:
18230:
18208:
18205:
18204:
18187:
18183:
18181:
18178:
18177:
18158:
18155:
18154:
18153:coordinates of
18138:
18135:
18134:
18093:
18083:
18079:
18078:
18074:
18063:
18059:
18058:
18054:
18046:
18042:
18041:
18037:
18029:
18025:
18016:
18012:
18006:
18002:
17998:
17994:
17988:
17975:
17971:
17970:
17957:
17944:
17940:
17939:
17929:
17916:
17912:
17911:
17895:
17891:
17876:
17872:
17863:
17859:
17857:
17854:
17853:
17830:
17825:
17824:
17809:
17804:
17803:
17801:
17798:
17797:
17780:
17776:
17767:
17763:
17761:
17758:
17757:
17755:polynomial maps
17734:
17730:
17724:
17720:
17709:
17706:
17705:
17685:
17682:
17681:
17659:
17655:
17646:
17642:
17636:
17632:
17631:
17627:
17622:
17619:
17618:
17584:
17581:
17580:
17564:
17561:
17560:
17538:
17534:
17525:
17521:
17515:
17511:
17510:
17506:
17495:
17492:
17491:
17471:
17468:
17467:
17466:is generic and
17451:
17448:
17447:
17422:
17417:
17416:
17414:
17411:
17410:
17408:
17392:
17389:
17388:
17372:
17369:
17368:
17346:
17343:
17342:
17340:
17324:
17321:
17320:
17296:
17293:
17292:
17272:
17269:
17268:
17250:
17247:
17246:
17244:
17217:
17212:
17211:
17196:
17191:
17190:
17182:
17179:
17178:
17159:
17155:
17146:
17142:
17140:
17137:
17136:
17134:
17110:
17107:
17106:
17087:
17084:
17083:
17077:
17041:
17037:
17035:
17032:
17031:
17029:
17023:
16995:
16971:
16967:
16962:
16959:
16958:
16952:
16938:
16932:
16926:
16920:
16892:
16868:
16864:
16859:
16856:
16855:
16849:
16839:
16820:
16806:
16800:
16794:
16739:
16735:
16721:
16718:
16717:
16694:
16690:
16681:
16677:
16662:
16658:
16649:
16645:
16630:
16626:
16617:
16613:
16598:
16594:
16585:
16581:
16576:
16573:
16572:
16566:
16546:
16542:
16537:
16534:
16533:
16525:
16518:
16512:
16506:
16500:
16497:
16491:
16453:
16440:
16426:
16420:
16400:
16396:
16390:
16386:
16374:
16370:
16364:
16360:
16358:
16355:
16354:
16335:
16331:
16322:
16318:
16316:
16313:
16312:
16310:
16304:
16278:
16274:
16269:
16266:
16265:
16263:
16257:
16238:
16234:
16232:
16229:
16228:
16226:
16203:
16199:
16197:
16194:
16193:
16176:
16172:
16170:
16167:
16166:
16160:
16138:
16135:
16134:
16117:
16113:
16104:
16100:
16098:
16095:
16094:
16066:
16062:
16053:
16049:
16047:
16044:
16043:
16041:
16017:
16013:
16004:
16000:
15999:
15995:
15977:
15973:
15968:
15965:
15964:
15931:
15927:
15911:
15907:
15898:
15894:
15882:
15876:
15873:
15872:
15837:
15833:
15831:
15828:
15827:
15821:
15815:
15784:
15773:
15765:
15760:
15753:
15752:
15748:
15743:
15740:
15739:
15726:
15720:
15707:
15683:
15680:
15679:
15677:
15671:
15665:
15643:
15640:
15639:
15636:
15602:
15598:
15593:
15590:
15589:
15559:
15556:
15555:
15553:
15525:
15522:
15521:
15470:
15466:
15461:
15458:
15457:
15444:)-bimodule and
15376:
15373:
15372:
15370:
15369:-module, where
15346:
15342:
15337:
15334:
15333:
15264:
15261:
15260:
15258:
15257:-module, where
15234:
15230:
15225:
15222:
15221:
15185:
15181:
15176:
15173:
15172:
15145:
15142:
15141:
15139:
15117:
15114:
15113:
15056:
15054:
15051:
15050:
15011:
15009:
15006:
15005:
14989:
14988:
14957:
14933:
14932:
14920:
14916:
14892:
14888:
14869:
14860:
14856:
14847:
14843:
14828:
14827:
14809:
14805:
14781:
14777:
14764:
14749:
14745:
14736:
14732:
14722:
14720:
14717:
14716:
14682:
14679:
14678:
14658:
14657:
14609:
14608:
14596:
14592:
14583:
14579:
14558:
14554:
14533:
14529:
14514:
14513:
14495:
14491:
14482:
14478:
14457:
14453:
14432:
14428:
14419:
14418:
14401:
14386:
14382:
14373:
14369:
14345:
14341:
14332:
14328:
14314:
14312:
14309:
14308:
14277:
14274:
14273:
14251:
14248:
14247:
14212:
14209:
14208:
14189:
14159:
14155:
14150:
14147:
14146:
14119:
14094:
14091:
14090:
14060:
14057:
14056:
14054:
14034:
14030:
14025:
14022:
14021:
14015:
13991:
13988:
13987:
13985:
13969:
13966:
13965:
13941:
13938:
13937:
13935:
13926:
13910:
13909:
13881:
13857:
13856:
13845:
13805:
13794:
13773:
13772:
13755:
13721:
13704:
13688:
13686:
13683:
13682:
13638:
13635:
13634:
13614:
13610:
13587:
13584:
13583:
13570:
13564:
13563:is imposed. If
13512:
13509:
13508:
13502:
13496:
13490:
13484:
13483:. In this case
13481:non-commutative
13474:these relations
13467:
13461:
13453:
13420:
13417:
13416:
13394:
13364:
13360:
13355:
13352:
13351:
13345:
13331:
13325:
13319:
13313:
13261:
13241:
13206:
13204:
13201:
13200:
13190:
13184:
13178:
13169:
13163:
13143:
13139:
13124:
13120:
13115:
13112:
13111:
13089:
13086:
13085:
13065:
13061:
13052:
13048:
13043:
13040:
13039:
12969:
12966:
12965:
12946:
12945:
12933:
12929:
12914:
12909:
12899:
12886:
12885:
12873:
12869:
12840:
12833:
12832:
12830:
12827:
12826:
12802:
12797:
12788:
12785:
12784:
12764:
12760:
12755:
12752:
12751:
12691:
12688:
12687:
12669:
12666:
12665:
12663:
12640:
12636:
12622:
12619:
12618:
12578:
12563:
12559:
12557:
12554:
12553:
12541:
12535:
12529:
12523:
12517:
12511:
12501:
12495:
12489:
12486:
12448:
12446:
12443:
12442:
12440:
12432:
12396:
12394:
12391:
12390:
12388:
12362:
12357:
12351:
12348:
12347:
12316:
12314:
12311:
12310:
12304:
12241:
12238:
12237:
12206:
12188:
12183:
12177:
12174:
12173:
12139:
12135:
12097:
12094:
12093:
12074:
12070:
12062:
12059:
12058:
12056:
12050:
12025:
12021:
12017:
12006:
11997:
11993:
11991:
11988:
11987:
11958:
11954:
11904:
11901:
11900:
11867:
11859:
11856:
11855:
11853:
11823:
11820:
11819:
11817:
11786:
11784:
11781:
11780:
11751:
11746:
11740:
11737:
11736:
11733:
11704:
11699:
11693:
11690:
11689:
11671:
11668:
11667:
11665:
11648:
11644:
11629:
11625:
11623:
11620:
11619:
11602:
11601:
11595:
11590:
11577:
11573:
11564:
11551:
11550:
11544:
11540:
11521:
11520:
11518:
11515:
11514:
11490:
11487:
11486:
11466:
11462:
11460:
11457:
11456:
11440:
11437:
11436:
11435:(The copies of
11401:
11390:
11368:
11363:
11357:
11354:
11353:
11338:
11308:
11305:
11304:
11302:
11260:
11257:
11256:
11233:
11229:
11221:
11218:
11217:
11215:evaluation map:
11210:
11207:
11172:
11170:
11169:
11163:
11159:
11153:
11149:
11140:
11138:
11137:
11128:
11124:
11110:
11107:
11106:
11089:
11085:
11079:
11077:
11076:
11070:
11066:
11057:
11055:
11054:
11048:
11046:
11045:
11039:
11024:
11020:
11019:
11017:
11014:
11013:
10994:
10990:
10988:
10985:
10984:
10982:
10954:
10951:
10950:
10944:
10925:
10920:
10914:
10911:
10910:
10908:
10904:
10898:
10868:
10864:
10849:
10845:
10833:
10829:
10817:
10813:
10812:
10808:
10800:
10796:
10787:
10783:
10777:
10773:
10772:
10768:
10751:
10747:
10738:
10734:
10728:
10724:
10723:
10719:
10705:
10702:
10701:
10682:
10677:
10665:
10662:
10661:
10659:
10642:
10637:
10625:
10622:
10621:
10615:
10609:
10599:
10593:
10555:
10550:
10544:
10541:
10540:
10523:
10519:
10517:
10514:
10513:
10503:
10497:
10475:
10471:
10459:
10446:
10442:
10427:
10408:
10404:
10395:
10391:
10386:
10383:
10382:
10365:
10361:
10359:
10356:
10355:
10349:
10332:
10328:
10326:
10323:
10322:
10316:
10281:
10274:
10265:
10258:
10232:
10231:
10222:
10221:
10211:
10207:
10192:
10187:
10181:
10178:
10177:
10164:
10158:
10152:
10128:
10124:
10122:
10119:
10118:
10095:
10085:
10081:
10077:
10076:
10064:
10060:
10051:
10039:
10035:
10020:
10016:
10015:
10013:
10003:
9981:
9979:
9960:
9955:
9949:
9946:
9945:
9939:
9908:
9905:
9904:
9898:
9892:
9889:
9883:
9881:General tensors
9817:
9814:
9813:
9811:
9776:
9775:
9763:
9759:
9747:
9743:
9741:
9729:
9725:
9713:
9709:
9707:
9695:
9691:
9679:
9675:
9673:
9661:
9657:
9645:
9641:
9638:
9637:
9625:
9621:
9609:
9605:
9603:
9591:
9587:
9575:
9571:
9569:
9557:
9553:
9541:
9537:
9535:
9523:
9519:
9507:
9503:
9500:
9499:
9487:
9483:
9471:
9467:
9465:
9453:
9449:
9437:
9433:
9431:
9419:
9415:
9403:
9399:
9397:
9385:
9381:
9369:
9365:
9362:
9361:
9349:
9345:
9333:
9329:
9327:
9315:
9311:
9299:
9295:
9293:
9281:
9277:
9265:
9261:
9259:
9247:
9243:
9231:
9227:
9220:
9219:
9209:
9208:
9201:
9200:
9188:
9184:
9182:
9170:
9166:
9163:
9162:
9150:
9146:
9144:
9132:
9128:
9121:
9120:
9108:
9104:
9102:
9095:
9094:
9082:
9078:
9076:
9064:
9060:
9057:
9056:
9044:
9040:
9038:
9026:
9022:
9015:
9014:
9002:
8998:
8995:
8994:
8987:
8986:
8974:
8970:
8968:
8956:
8952:
8949:
8948:
8936:
8932:
8930:
8918:
8914:
8907:
8906:
8894:
8890:
8888:
8881:
8880:
8868:
8864:
8862:
8850:
8846:
8843:
8842:
8830:
8826:
8824:
8812:
8808:
8801:
8800:
8788:
8784:
8777:
8776:
8766:
8765:
8753:
8749:
8747:
8735:
8731:
8728:
8727:
8715:
8711:
8709:
8697:
8693:
8686:
8685:
8675:
8674:
8662:
8658:
8656:
8644:
8640:
8637:
8636:
8624:
8620:
8618:
8606:
8602:
8595:
8594:
8592:
8589:
8588:
8567:
8566:
8554:
8550:
8548:
8536:
8532:
8529:
8528:
8516:
8512:
8510:
8498:
8494:
8487:
8486:
8469:
8468:
8456:
8452:
8450:
8438:
8434:
8431:
8430:
8418:
8414:
8412:
8400:
8396:
8389:
8388:
8380:
8377:
8376:
8370:
8364:
8358:
8344:
8318:
8315:
8314:
8292:
8289:
8288:
8278:
8272:
8241:
8237:
8223:category theory
8122:
8119:
8118:
8036:
8033:
8032:
7980:
7977:
7976:
7943:
7940:
7939:
7937:
7909:
7906:
7905:
7880:
7877:
7876:
7806:
7803:
7802:
7750:
7747:
7746:
7741:tensor product:
7736:
7706:
7703:
7702:
7700:
7697:
7690:
7663:
7659:
7647:
7643:
7641:
7638:
7637:
7635:
7606:
7602:
7578:
7574:
7565:
7561:
7546:
7542:
7540:
7537:
7536:
7529:
7525:
7518:
7514:
7494:
7490:
7488:
7485:
7484:
7474:
7473:
7446:
7443:
7442:
7408:
7405:
7404:
7365:
7362:
7361:
7359:
7335:
7332:
7331:
7329:
7302:
7299:
7298:
7296:
7274:
7271:
7270:
7230:
7227:
7226:
7203:
7200:
7199:
7183:
7180:
7179:
7176:
7132:
7129:
7128:
7126:
7092:
7089:
7088:
7024:
7021:
7020:
6987:
6984:
6983:
6981:
6974:
6963:
6957:
6935:
6932:
6931:
6922:
6916:
6894:
6891:
6890:
6880:
6874:
6871:
6866:
6843:
6840:
6839:
6837:
6821:
6818:
6817:
6803:
6802:
6791:
6783:
6763:
6762:
6757:
6749:
6737:
6732:
6730:
6727:
6726:
6703:
6698:
6697:
6695:
6692:
6691:
6641:
6637:
6623:
6620:
6619:
6602:
6597:
6596:
6588:
6585:
6584:
6567:
6562:
6561:
6553:
6550:
6549:
6492:
6489:
6488:
6486:
6463:
6458:
6457:
6443:
6440:
6439:
6420:
6415:
6414:
6406:
6403:
6402:
6400:
6383:
6378:
6377:
6369:
6366:
6365:
6349:
6346:
6345:
6329:
6326:
6325:
6289:
6286:
6285:
6283:
6264:
6261:
6260:
6214:
6211:
6210:
6188:
6185:
6184:
6168:
6165:
6164:
6147:
6142:
6141:
6139:
6136:
6135:
6090:
6087:
6086:
6084:
6039:
6036:
6035:
6017:
6014:
6013:
5997:
5994:
5993:
5974:
5969:
5968:
5960:
5957:
5956:
5954:
5937:
5932:
5931:
5923:
5920:
5919:
5881:
5858:
5856:
5855:
5844:
5840:
5834:
5830:
5829:
5825:
5824:
5822:
5819:
5818:
5791:
5787:
5772:
5768:
5767:
5763:
5749:
5745:
5730:
5726:
5725:
5721:
5720:
5716:
5696:
5693:
5692:
5672:
5667:
5666:
5657:
5652:
5651:
5642:
5637:
5636:
5628:
5625:
5624:
5604:
5599:
5598:
5590:
5587:
5586:
5570:
5567:
5566:
5550:
5547:
5546:
5543:
5479:
5475:
5466:
5462:
5461:
5457:
5453:
5449:
5447:
5444:
5443:
5425:
5422:
5421:
5419:
5402:
5398:
5383:
5379:
5377:
5374:
5373:
5357:
5354:
5353:
5336:
5332:
5317:
5313:
5311:
5308:
5307:
5291:
5288:
5287:
5271:
5268:
5267:
5251:
5248:
5247:
5225:
5222:
5221:
5219:
5202:
5198:
5196:
5193:
5192:
5175:
5171:
5156:
5152:
5150:
5147:
5146:
5125:
5122:
5121:
5119:
5102:
5098:
5096:
5093:
5092:
5071:
5067:
5052:
5048:
5046:
5043:
5042:
5008:
5004:
4995:
4991:
4990:
4986:
4977:
4966:
4960:
4957:
4956:
4954:
4931:
4927:
4912:
4908:
4906:
4903:
4902:
4879:
4875:
4860:
4856:
4854:
4851:
4850:
4834:
4831:
4830:
4812:
4809:
4808:
4791:
4788:
4787:
4771:
4768:
4767:
4724:
4721:
4720:
4718:
4702:
4699:
4698:
4682:
4679:
4678:
4662:
4659:
4658:
4642:
4639:
4638:
4610:
4607:
4606:
4572:
4569:
4568:
4552:
4549:
4548:
4524:
4521:
4520:
4518:
4514:
4505:
4475:
4472:
4471:
4469:
4447:
4444:
4443:
4406:
4405:
4382:
4379:
4378:
4362:
4348:
4347:
4339:
4336:
4335:
4293:
4292:
4290:
4287:
4286:
4284:
4244:
4241:
4240:
4238:
4214:
4211:
4210:
4208:
4186:
4183:
4182:
4136:
4134:
4131:
4130:
4106:
4103:
4102:
4100:
4096:
4092:
4041:
4038:
4037:
4036:
4031:
4018:
4015:
4014:
4013:
4012:
4006:
3999:
3959:
3956:
3955:
3953:
3925:
3922:
3921:
3896:
3879:
3876:
3875:
3841:
3838:
3837:
3835:
3812:
3808:
3799:
3795:
3787:
3784:
3783:
3758:
3754:
3745:
3741:
3733:
3730:
3729:
3727:
3710:
3709:
3681:
3660:
3659:
3631:
3610:
3609:
3597:
3593:
3572:
3568:
3552:
3543:
3539:
3530:
3526:
3514:
3513:
3495:
3491:
3470:
3466:
3456:
3441:
3437:
3428:
3424:
3417:
3415:
3412:
3411:
3400:
3396:
3393:linear subspace
3388:
3381:
3353:
3350:
3349:
3345:
3317:
3314:
3313:
3285:
3282:
3281:
3254:
3251:
3250:
3248:
3226:
3223:
3222:
3194:
3191:
3190:
3183:
3157:
3154:
3153:
3146:
3138:
3128:
3124:
3118:
3078:
3075:
3074:
3072:
3056:
3053:
3052:
3026:
3023:
3022:
3010:satisfying the
2989:
2986:
2985:
2963:
2960:
2959:
2913:
2911:
2908:
2907:
2888:
2884:
2882:
2879:
2878:
2876:
2859:
2855:
2853:
2850:
2849:
2818:
2815:
2814:
2783:
2780:
2779:
2763:
2762:
2743:
2739:
2733:
2729:
2721:
2717:
2710:
2698:
2694:
2687:
2674:
2673:
2667:
2666:
2656:
2652:
2644:
2640:
2633:
2623:
2622:
2613:
2612:
2602:
2598:
2590:
2586:
2579:
2569:
2568:
2561:
2548:
2546:
2543:
2542:
2520:
2517:
2516:
2494:
2491:
2490:
2464:
2460:
2451:
2447:
2445:
2442:
2441:
2439:
2423:
2420:
2419:
2391:
2388:
2387:
2385:
2348:
2346:
2343:
2342:
2286:
2282:
2275:
2263:
2259:
2252:
2240:
2237:
2236:
2214:
2211:
2210:
2194:
2191:
2190:
2168:
2164:
2157:
2153:
2139:
2128:
2099:
2098:
2094:
2084:
2083:
2079:
2071:
2067:
2056:
2055:
2043:
2039:
2028:
2027:
1988:
1985:
1984:
1944:
1941:
1940:
1923:
1919:
1910:
1906:
1904:
1901:
1900:
1878:
1875:
1874:
1855:
1851:
1842:
1838:
1836:
1833:
1832:
1830:
1790:
1787:
1786:
1770:
1767:
1766:
1728:
1724:
1718:
1714:
1706:
1702:
1695:
1683:
1679:
1672:
1645:
1642:
1641:
1625:
1622:
1621:
1604:
1600:
1598:
1595:
1594:
1577:
1573:
1571:
1568:
1567:
1543:
1539:
1531:
1527:
1520:
1504:
1493:
1489:
1481:
1477:
1470:
1458:
1455:
1454:
1437:
1433:
1431:
1428:
1427:
1410:
1406:
1404:
1401:
1400:
1384:
1381:
1380:
1364:
1361:
1360:
1324:
1321:
1320:
1318:
1278:
1275:
1274:
1255:
1251:
1242:
1238:
1236:
1233:
1232:
1230:
1208:
1205:
1204:
1178:
1175:
1174:
1152:
1148:
1146:
1143:
1142:
1140:
1123:
1119:
1117:
1114:
1113:
1085:
1082:
1081:
1079:
1059:
1055:
1040:
1036:
1013:
1010:
1009:
982:
979:
978:
976:
972:
955:
951:
942:
938:
936:
933:
932:
928:
900:
897:
896:
874:
871:
870:
862:
845:
841:
832:
828:
826:
823:
822:
793:
790:
789:
770:
766:
758:
755:
754:
752:
735:
731:
723:
720:
719:
697:
694:
693:
689:
685:
663:
660:
659:
634:
630:
628:
625:
624:
622:
605:
601:
599:
596:
595:
588:
577:
573:
570:
534:
526:cotangent space
463:
460:
459:
452:
430:
427:
426:
419:
415:
393:
390:
389:
383:
377:
351:
348:
347:
325:
322:
321:
284:
281:
280:
276:
272:
246:
243:
242:
215:
212:
211:
209:
187:
184:
183:
128:
125:
124:
90:
87:
86:
78:(over the same
73:
67:
42:
39:
38:
28:
17:
12:
11:
5:
21320:
21310:
21309:
21304:
21299:
21282:
21281:
21279:
21278:
21273:
21271:Woldemar Voigt
21268:
21263:
21258:
21253:
21248:
21243:
21238:
21236:Leonhard Euler
21233:
21228:
21223:
21218:
21212:
21210:
21208:Mathematicians
21204:
21203:
21200:
21199:
21197:
21196:
21191:
21186:
21181:
21176:
21171:
21166:
21161:
21156:
21150:
21148:
21144:
21143:
21141:
21140:
21135:
21133:Torsion tensor
21130:
21125:
21120:
21115:
21110:
21105:
21099:
21097:
21090:
21086:
21085:
21083:
21082:
21077:
21072:
21067:
21062:
21057:
21052:
21047:
21042:
21037:
21032:
21027:
21022:
21017:
21012:
21007:
21002:
20997:
20992:
20986:
20984:
20978:
20977:
20975:
20974:
20968:
20966:Tensor product
20963:
20958:
20956:Symmetrization
20953:
20948:
20946:Lie derivative
20943:
20938:
20933:
20928:
20923:
20917:
20915:
20909:
20908:
20906:
20905:
20900:
20895:
20890:
20885:
20880:
20875:
20870:
20868:Tensor density
20865:
20860:
20854:
20852:
20846:
20845:
20843:
20842:
20840:Voigt notation
20837:
20832:
20827:
20825:Ricci calculus
20822:
20817:
20812:
20810:Index notation
20807:
20802:
20796:
20794:
20790:
20789:
20786:
20785:
20783:
20782:
20777:
20772:
20767:
20762:
20756:
20754:
20752:
20751:
20746:
20740:
20737:
20736:
20734:
20733:
20728:
20726:Tensor algebra
20723:
20718:
20713:
20708:
20706:Dyadic algebra
20703:
20698:
20692:
20690:
20681:
20677:
20676:
20669:
20666:
20665:
20658:
20657:
20650:
20643:
20635:
20629:
20628:
20620:
20606:
20590:
20584:
20571:
20565:
20545:
20524:
20504:
20498:
20482:
20476:
20460:
20455:978-0387715674
20454:
20441:
20425:
20419:
20401:
20398:
20395:
20394:
20387:
20369:
20354:
20328:
20310:
20289:
20246:
20239:
20221:
20208:
20194:. 2008-11-13.
20179:
20162:
20142:
20135:
20117:
20102:
20086:
20085:
20083:
20080:
20079:
20078:
20072:
20066:
20063:Tensor algebra
20060:
20054:
20049:
20041:
20038:
20032:(for example,
20011:differentiable
19952:
19949:
19947:
19944:
19925:
19922:
19919:
19914:
19910:
19906:
19901:
19898:
19895:
19891:
19887:
19884:
19881:
19878:
19875:
19870:
19867:
19864:
19860:
19856:
19851:
19847:
19843:
19840:
19837:
19832:
19825:
19819:
19815:
19812:
19809:
19806:
19803:
19796:
19793:
19790:
19785:
19781:
19758:
19753:
19746:
19742:
19738:
19735:
19730:
19726:
19722:
19717:
19713:
19709:
19706:
19701:
19697:
19693:
19688:
19684:
19680:
19675:
19671:
19667:
19662:
19658:
19652:
19645:
19640:
19637:
19634:
19631:
19628:
19625:
19622:
19588:exterior power
19567:
19562:
19558:
19529:
19526:
19523:
19520:
19517:
19493:
19489:
19485:
19480:
19476:
19472:
19469:
19464:
19460:
19456:
19451:
19447:
19422:
19418:
19414:
19409:
19405:
19382:
19378:
19374:
19369:
19365:
19342:
19337:
19330:
19326:
19322:
19319:
19314:
19310:
19306:
19301:
19297:
19293:
19290:
19285:
19281:
19277:
19272:
19268:
19264:
19259:
19255:
19251:
19246:
19242:
19236:
19229:
19224:
19221:
19218:
19215:
19212:
19209:
19206:
19178:
19175:
19172:
19169:
19166:
19163:
19160:
19157:
19154:
19151:
19146:
19141:
19138:
19135:
19132:
19129:
19126:
19123:
19103:
19100:
19097:
19049:tensor algebra
19044:
19041:
19032:
19029:
19001:Main article:
18998:
18995:
18990:Main article:
18987:
18984:
18973:Main article:
18970:
18967:
18962:Main article:
18959:
18956:
18929:
18926:
18921:
18918:
18915:
18911:
18907:
18904:
18901:
18896:
18893:
18890:
18886:
18882:
18879:
18876:
18871:
18867:
18863:
18860:
18857:
18852:
18848:
18844:
18841:
18838:
18835:
18830:
18827:
18824:
18820:
18816:
18813:
18810:
18805:
18801:
18797:
18794:
18791:
18788:
18785:
18782:
18762:
18742:
18722:
18717:
18713:
18709:
18706:
18703:
18698:
18694:
18690:
18687:
18667:
18662:
18658:
18654:
18651:
18648:
18643:
18639:
18635:
18632:
18617:
18614:
18609:Main article:
18606:
18603:
18587:Main article:
18584:
18581:
18569:Main article:
18566:
18563:
18535:Hilbert spaces
18521:
18518:
18516:
18513:
18497:
18494:
18491:
18467:
18464:
18459:
18453:
18449:
18445:
18440:
18436:
18432:
18430:
18427:
18425:
18421:
18417:
18412:
18408:
18404:
18402:
18398:
18394:
18389:
18385:
18381:
18380:
18375:
18371:
18367:
18365:
18362:
18358:
18354:
18350:
18346:
18342:
18338:
18337:
18335:
18306:
18283:
18277:
18273:
18269:
18266:
18263:
18258:
18254:
18249:
18245:
18241:
18218:
18215:
18212:
18190:
18186:
18162:
18142:
18120:
18117:
18114:
18111:
18108:
18105:
18102:
18086:
18082:
18077:
18073:
18066:
18062:
18057:
18049:
18045:
18040:
18032:
18028:
18024:
18019:
18015:
18009:
18005:
18001:
17997:
17991:
17986:
17983:
17978:
17974:
17969:
17965:
17960:
17955:
17952:
17947:
17943:
17938:
17932:
17927:
17924:
17919:
17915:
17910:
17906:
17903:
17898:
17894:
17890:
17887:
17884:
17879:
17875:
17871:
17866:
17862:
17839:
17836:
17833:
17828:
17823:
17818:
17815:
17812:
17807:
17783:
17779:
17775:
17770:
17766:
17740:
17737:
17733:
17727:
17723:
17719:
17716:
17713:
17702:characteristic
17689:
17669:
17662:
17658:
17654:
17649:
17645:
17639:
17635:
17630:
17626:
17606:
17603:
17600:
17597:
17594:
17591:
17588:
17568:
17548:
17541:
17537:
17533:
17528:
17524:
17518:
17514:
17509:
17505:
17502:
17499:
17475:
17455:
17431:
17428:
17425:
17420:
17396:
17376:
17350:
17328:
17300:
17276:
17254:
17226:
17223:
17220:
17215:
17210:
17205:
17202:
17199:
17194:
17189:
17186:
17162:
17158:
17154:
17149:
17145:
17114:
17091:
17076:
17073:
17056:
17053:
17050:
17047:
17044:
17040:
17011:
17008:
17005:
17002:
16998:
16994:
16991:
16988:
16985:
16982:
16979:
16974:
16970:
16966:
16908:
16905:
16902:
16899:
16895:
16891:
16888:
16885:
16882:
16879:
16876:
16871:
16867:
16863:
16780:
16777:
16774:
16771:
16768:
16765:
16762:
16759:
16756:
16753:
16750:
16747:
16742:
16738:
16734:
16731:
16728:
16725:
16705:
16702:
16697:
16693:
16689:
16684:
16680:
16676:
16673:
16670:
16665:
16661:
16657:
16652:
16648:
16644:
16641:
16638:
16633:
16629:
16625:
16620:
16616:
16612:
16609:
16606:
16601:
16597:
16593:
16588:
16584:
16580:
16554:
16549:
16545:
16541:
16493:Main article:
16490:
16487:
16408:
16403:
16399:
16393:
16389:
16385:
16382:
16377:
16373:
16367:
16363:
16338:
16334:
16330:
16325:
16321:
16286:
16281:
16277:
16273:
16241:
16237:
16214:
16209:
16206:
16202:
16179:
16175:
16148:
16145:
16142:
16120:
16116:
16112:
16107:
16103:
16093:, and the map
16080:
16075:
16072:
16069:
16065:
16061:
16056:
16052:
16026:
16020:
16016:
16012:
16007:
16003:
15998:
15994:
15991:
15988:
15985:
15980:
15976:
15972:
15948:
15945:
15942:
15937:
15934:
15930:
15925:
15922:
15919:
15914:
15910:
15904:
15901:
15897:
15891:
15888:
15885:
15881:
15860:
15857:
15854:
15851:
15848:
15845:
15840:
15836:
15797:
15794:
15791:
15787:
15783:
15780:
15776:
15772:
15768:
15763:
15756:
15751:
15747:
15693:
15690:
15687:
15653:
15650:
15647:
15635:
15632:
15631:
15630:
15610:
15605:
15601:
15597:
15575:
15572:
15569:
15566:
15563:
15541:
15538:
15535:
15532:
15529:
15498:
15478:
15473:
15469:
15465:
15430:
15416:
15413:
15410:
15407:
15404:
15401:
15398:
15395:
15392:
15389:
15386:
15383:
15380:
15354:
15349:
15345:
15341:
15318:
15304:
15301:
15298:
15295:
15292:
15289:
15286:
15283:
15280:
15277:
15274:
15271:
15268:
15242:
15237:
15233:
15229:
15193:
15188:
15184:
15180:
15155:
15152:
15149:
15127:
15124:
15121:
15101:
15098:
15095:
15092:
15089:
15086:
15083:
15080:
15077:
15074:
15071:
15068:
15063:
15060:
15038:
15035:
15032:
15029:
15026:
15023:
15018:
15015:
14987:
14984:
14981:
14978:
14975:
14972:
14969:
14966:
14963:
14960:
14958:
14956:
14953:
14950:
14947:
14944:
14941:
14938:
14935:
14934:
14931:
14928:
14923:
14919:
14915:
14912:
14909:
14906:
14903:
14900:
14895:
14891:
14887:
14884:
14881:
14878:
14875:
14872:
14870:
14868:
14863:
14859:
14855:
14850:
14846:
14842:
14839:
14836:
14833:
14830:
14829:
14826:
14823:
14820:
14817:
14812:
14808:
14804:
14801:
14798:
14795:
14792:
14789:
14784:
14780:
14776:
14773:
14770:
14767:
14765:
14763:
14760:
14757:
14752:
14748:
14744:
14739:
14735:
14731:
14728:
14725:
14724:
14704:
14701:
14698:
14695:
14692:
14689:
14686:
14656:
14653:
14650:
14647:
14644:
14641:
14638:
14635:
14632:
14629:
14626:
14623:
14620:
14617:
14614:
14612:
14610:
14607:
14604:
14599:
14595:
14591:
14586:
14582:
14578:
14575:
14572:
14569:
14566:
14561:
14557:
14553:
14550:
14547:
14544:
14541:
14536:
14532:
14528:
14525:
14522:
14519:
14517:
14515:
14512:
14509:
14506:
14503:
14498:
14494:
14490:
14485:
14481:
14477:
14474:
14471:
14468:
14465:
14460:
14456:
14452:
14449:
14446:
14443:
14440:
14435:
14431:
14427:
14424:
14422:
14420:
14417:
14414:
14411:
14408:
14400:
14397:
14394:
14389:
14385:
14381:
14376:
14372:
14368:
14365:
14362:
14359:
14356:
14353:
14348:
14344:
14340:
14335:
14331:
14327:
14324:
14321:
14318:
14316:
14296:
14293:
14290:
14287:
14284:
14281:
14261:
14258:
14255:
14231:
14228:
14225:
14222:
14219:
14216:
14196:
14192:
14188:
14185:
14182:
14179:
14176:
14173:
14170:
14167:
14162:
14158:
14154:
14118:
14115:
14098:
14076:
14073:
14070:
14067:
14064:
14042:
14037:
14033:
14029:
14001:
13998:
13995:
13973:
13951:
13948:
13945:
13908:
13905:
13902:
13899:
13896:
13893:
13890:
13887:
13884:
13882:
13880:
13877:
13874:
13871:
13868:
13865:
13862:
13859:
13858:
13855:
13851:
13848:
13844:
13841:
13838:
13835:
13832:
13829:
13826:
13823:
13820:
13817:
13814:
13811:
13808:
13806:
13804:
13800:
13797:
13793:
13790:
13787:
13784:
13781:
13778:
13775:
13774:
13771:
13768:
13765:
13761:
13758:
13754:
13751:
13748:
13745:
13742:
13739:
13736:
13733:
13730:
13727:
13724:
13722:
13720:
13717:
13714:
13710:
13707:
13703:
13700:
13697:
13694:
13691:
13690:
13666:
13663:
13660:
13657:
13654:
13651:
13648:
13645:
13642:
13622:
13617:
13613:
13609:
13606:
13603:
13600:
13597:
13594:
13591:
13552:
13549:
13546:
13543:
13540:
13537:
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13531:
13528:
13525:
13522:
13519:
13516:
13439:
13436:
13433:
13430:
13427:
13424:
13404:
13401:
13397:
13393:
13390:
13387:
13384:
13381:
13378:
13375:
13372:
13367:
13363:
13359:
13315:Main article:
13312:
13309:
13292:
13289:
13286:
13283:
13280:
13277:
13274:
13270:
13267:
13264:
13260:
13257:
13254:
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13240:
13237:
13234:
13231:
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13225:
13222:
13219:
13215:
13212:
13209:
13151:
13146:
13142:
13138:
13135:
13132:
13127:
13123:
13119:
13099:
13096:
13093:
13073:
13068:
13064:
13060:
13055:
13051:
13047:
13027:
13024:
13021:
13018:
13015:
13012:
13009:
13006:
13003:
13000:
12997:
12994:
12991:
12988:
12985:
12982:
12979:
12976:
12973:
12949:
12944:
12941:
12936:
12932:
12928:
12925:
12922:
12917:
12912:
12908:
12902:
12898:
12894:
12891:
12888:
12887:
12884:
12881:
12876:
12872:
12868:
12865:
12862:
12859:
12856:
12853:
12849:
12846:
12843:
12839:
12838:
12836:
12810:
12805:
12800:
12796:
12792:
12772:
12767:
12763:
12759:
12737:
12734:
12731:
12728:
12725:
12722:
12719:
12716:
12713:
12710:
12707:
12704:
12701:
12698:
12695:
12673:
12651:
12648:
12643:
12639:
12635:
12632:
12629:
12626:
12606:
12603:
12600:
12597:
12594:
12591:
12587:
12584:
12581:
12577:
12574:
12571:
12566:
12562:
12485:
12482:
12467:
12464:
12461:
12457:
12454:
12451:
12415:
12412:
12409:
12405:
12402:
12399:
12376:
12373:
12370:
12365:
12360:
12356:
12335:
12332:
12329:
12325:
12322:
12319:
12290:
12287:
12284:
12281:
12278:
12275:
12272:
12269:
12266:
12263:
12260:
12257:
12254:
12251:
12248:
12245:
12225:
12222:
12219:
12215:
12212:
12209:
12205:
12202:
12199:
12196:
12191:
12186:
12182:
12159:
12156:
12153:
12150:
12147:
12142:
12138:
12134:
12131:
12128:
12125:
12122:
12119:
12116:
12113:
12110:
12107:
12104:
12101:
12077:
12073:
12069:
12066:
12033:
12028:
12024:
12020:
12015:
12012:
12009:
12005:
12000:
11996:
11975:
11972:
11969:
11966:
11961:
11957:
11953:
11950:
11947:
11944:
11941:
11938:
11935:
11932:
11929:
11926:
11923:
11920:
11917:
11914:
11911:
11908:
11886:
11883:
11880:
11876:
11873:
11870:
11866:
11863:
11839:
11836:
11833:
11830:
11827:
11805:
11802:
11799:
11795:
11792:
11789:
11765:
11762:
11759:
11754:
11749:
11745:
11732:
11729:
11707:
11702:
11698:
11675:
11651:
11647:
11643:
11640:
11637:
11632:
11628:
11605:
11598:
11593:
11589:
11585:
11580:
11576:
11572:
11567:
11563:
11559:
11556:
11553:
11552:
11547:
11543:
11539:
11536:
11533:
11530:
11527:
11526:
11524:
11508:
11494:
11469:
11465:
11444:
11424:
11421:
11418:
11415:
11410:
11407:
11404:
11399:
11396:
11393:
11389:
11385:
11382:
11379:
11376:
11371:
11366:
11362:
11324:
11321:
11318:
11315:
11312:
11288:
11285:
11282:
11279:
11276:
11273:
11270:
11267:
11264:
11244:
11241:
11236:
11232:
11228:
11225:
11206:
11203:
11199:tensor algebra
11180:
11175:
11166:
11162:
11156:
11152:
11148:
11143:
11134:
11131:
11127:
11123:
11120:
11117:
11114:
11092:
11088:
11082:
11073:
11069:
11065:
11060:
11051:
11042:
11037:
11033:
11030:
11027:
11023:
10997:
10993:
10970:
10967:
10964:
10961:
10958:
10928:
10923:
10919:
10884:
10877:
10874:
10871:
10867:
10863:
10858:
10855:
10852:
10848:
10842:
10839:
10836:
10832:
10826:
10823:
10820:
10816:
10811:
10803:
10799:
10795:
10790:
10786:
10780:
10776:
10771:
10767:
10760:
10757:
10754:
10750:
10746:
10741:
10737:
10731:
10727:
10722:
10718:
10715:
10712:
10709:
10685:
10680:
10676:
10672:
10669:
10645:
10640:
10636:
10632:
10629:
10569:
10566:
10563:
10558:
10553:
10549:
10526:
10522:
10483:
10478:
10474:
10470:
10466:
10462:
10458:
10454:
10449:
10445:
10441:
10438:
10434:
10430:
10426:
10422:
10419:
10416:
10411:
10407:
10403:
10398:
10394:
10390:
10368:
10364:
10335:
10331:
10302:
10299:
10296:
10293:
10287:
10284:
10280:
10277:
10271:
10268:
10264:
10261:
10257:
10253:
10250:
10247:
10244:
10238:
10235:
10228:
10225:
10220:
10214:
10210:
10206:
10203:
10200:
10195:
10190:
10186:
10175:
10131:
10127:
10106:
10101:
10098:
10093:
10088:
10084:
10080:
10075:
10070:
10067:
10063:
10059:
10054:
10048:
10042:
10038:
10034:
10031:
10028:
10023:
10019:
10011:
10006:
10000:
9996:
9993:
9990:
9987:
9984:
9977:
9974:
9971:
9968:
9963:
9958:
9954:
9924:
9921:
9918:
9915:
9912:
9882:
9879:
9875:dyadic product
9857:
9854:
9851:
9848:
9845:
9842:
9839:
9836:
9833:
9830:
9827:
9824:
9821:
9801:
9785:
9780:
9772:
9769:
9766:
9762:
9756:
9753:
9750:
9746:
9742:
9738:
9735:
9732:
9728:
9722:
9719:
9716:
9712:
9708:
9704:
9701:
9698:
9694:
9688:
9685:
9682:
9678:
9674:
9670:
9667:
9664:
9660:
9654:
9651:
9648:
9644:
9640:
9639:
9634:
9631:
9628:
9624:
9618:
9615:
9612:
9608:
9604:
9600:
9597:
9594:
9590:
9584:
9581:
9578:
9574:
9570:
9566:
9563:
9560:
9556:
9550:
9547:
9544:
9540:
9536:
9532:
9529:
9526:
9522:
9516:
9513:
9510:
9506:
9502:
9501:
9496:
9493:
9490:
9486:
9480:
9477:
9474:
9470:
9466:
9462:
9459:
9456:
9452:
9446:
9443:
9440:
9436:
9432:
9428:
9425:
9422:
9418:
9412:
9409:
9406:
9402:
9398:
9394:
9391:
9388:
9384:
9378:
9375:
9372:
9368:
9364:
9363:
9358:
9355:
9352:
9348:
9342:
9339:
9336:
9332:
9328:
9324:
9321:
9318:
9314:
9308:
9305:
9302:
9298:
9294:
9290:
9287:
9284:
9280:
9274:
9271:
9268:
9264:
9260:
9256:
9253:
9250:
9246:
9240:
9237:
9234:
9230:
9226:
9225:
9223:
9218:
9213:
9205:
9197:
9194:
9191:
9187:
9183:
9179:
9176:
9173:
9169:
9165:
9164:
9159:
9156:
9153:
9149:
9145:
9141:
9138:
9135:
9131:
9127:
9126:
9124:
9117:
9114:
9111:
9107:
9103:
9099:
9091:
9088:
9085:
9081:
9077:
9073:
9070:
9067:
9063:
9059:
9058:
9053:
9050:
9047:
9043:
9039:
9035:
9032:
9029:
9025:
9021:
9020:
9018:
9011:
9008:
9005:
9001:
8997:
8996:
8991:
8983:
8980:
8977:
8973:
8969:
8965:
8962:
8959:
8955:
8951:
8950:
8945:
8942:
8939:
8935:
8931:
8927:
8924:
8921:
8917:
8913:
8912:
8910:
8903:
8900:
8897:
8893:
8889:
8885:
8877:
8874:
8871:
8867:
8863:
8859:
8856:
8853:
8849:
8845:
8844:
8839:
8836:
8833:
8829:
8825:
8821:
8818:
8815:
8811:
8807:
8806:
8804:
8797:
8794:
8791:
8787:
8783:
8782:
8780:
8775:
8770:
8762:
8759:
8756:
8752:
8748:
8744:
8741:
8738:
8734:
8730:
8729:
8724:
8721:
8718:
8714:
8710:
8706:
8703:
8700:
8696:
8692:
8691:
8689:
8684:
8679:
8671:
8668:
8665:
8661:
8657:
8653:
8650:
8647:
8643:
8639:
8638:
8633:
8630:
8627:
8623:
8619:
8615:
8612:
8609:
8605:
8601:
8600:
8598:
8576:
8571:
8563:
8560:
8557:
8553:
8549:
8545:
8542:
8539:
8535:
8531:
8530:
8525:
8522:
8519:
8515:
8511:
8507:
8504:
8501:
8497:
8493:
8492:
8490:
8485:
8482:
8478:
8473:
8465:
8462:
8459:
8455:
8451:
8447:
8444:
8441:
8437:
8433:
8432:
8427:
8424:
8421:
8417:
8413:
8409:
8406:
8403:
8399:
8395:
8394:
8392:
8387:
8384:
8328:
8325:
8322:
8302:
8299:
8296:
8258:exact sequence
8219:
8218:
8207:
8204:
8201:
8198:
8195:
8192:
8189:
8186:
8183:
8180:
8177:
8174:
8171:
8168:
8165:
8162:
8159:
8156:
8153:
8150:
8147:
8144:
8141:
8138:
8135:
8132:
8129:
8126:
8112:
8111:
8100:
8097:
8094:
8091:
8088:
8085:
8082:
8079:
8076:
8073:
8070:
8067:
8064:
8061:
8058:
8055:
8052:
8049:
8046:
8043:
8040:
8026:
8025:
8014:
8011:
8008:
8005:
8002:
7999:
7996:
7993:
7990:
7987:
7984:
7959:
7956:
7953:
7950:
7947:
7925:
7922:
7919:
7916:
7913:
7890:
7887:
7884:
7873:
7872:
7861:
7858:
7855:
7852:
7849:
7846:
7843:
7840:
7837:
7834:
7831:
7828:
7825:
7822:
7819:
7816:
7813:
7810:
7796:
7795:
7784:
7781:
7778:
7775:
7772:
7769:
7766:
7763:
7760:
7757:
7754:
7722:
7719:
7716:
7713:
7710:
7689:
7686:
7669:
7666:
7662:
7658:
7653:
7650:
7646:
7632:
7631:
7618:
7615:
7612:
7609:
7605:
7601:
7598:
7595:
7590:
7587:
7584:
7581:
7577:
7573:
7568:
7564:
7560:
7557:
7554:
7549:
7545:
7500:
7497:
7493:
7481:tensor algebra
7456:
7453:
7450:
7430:
7427:
7424:
7421:
7418:
7415:
7412:
7400:, in general.
7387:
7384:
7381:
7378:
7375:
7372:
7369:
7345:
7342:
7339:
7312:
7309:
7306:
7284:
7281:
7278:
7267:
7266:
7255:
7252:
7249:
7246:
7243:
7240:
7237:
7234:
7207:
7187:
7175:
7172:
7154:
7151:
7148:
7145:
7142:
7139:
7136:
7114:
7111:
7108:
7105:
7102:
7099:
7096:
7085:
7084:
7073:
7070:
7067:
7064:
7061:
7058:
7055:
7052:
7049:
7046:
7043:
7040:
7037:
7034:
7031:
7028:
7003:
7000:
6997:
6994:
6991:
6973:
6970:
6945:
6942:
6939:
6904:
6901:
6898:
6870:
6867:
6865:
6862:
6847:
6825:
6801:
6798:
6795:
6792:
6789:
6785:
6782:
6779:
6776:
6773:
6770:
6767:
6764:
6761:
6758:
6755:
6751:
6748:
6745:
6742:
6739:
6735:
6734:
6712:
6709:
6706:
6701:
6678:
6674:
6671:
6668:
6665:
6662:
6659:
6656:
6653:
6650:
6647:
6644:
6640:
6636:
6633:
6630:
6627:
6605:
6600:
6595:
6592:
6570:
6565:
6560:
6557:
6535:
6532:
6529:
6526:
6523:
6520:
6517:
6514:
6511:
6508:
6505:
6502:
6499:
6496:
6472:
6469:
6466:
6461:
6456:
6453:
6450:
6447:
6423:
6418:
6413:
6410:
6386:
6381:
6376:
6373:
6353:
6333:
6311:
6308:
6305:
6302:
6299:
6296:
6293:
6271:
6268:
6248:
6245:
6242:
6239:
6236:
6233:
6230:
6227:
6224:
6221:
6218:
6198:
6195:
6192:
6172:
6150:
6145:
6118:
6115:
6112:
6109:
6106:
6103:
6100:
6097:
6094:
6072:
6069:
6066:
6063:
6060:
6057:
6053:
6049:
6046:
6043:
6022:
6001:
5977:
5972:
5967:
5964:
5940:
5935:
5930:
5927:
5902:
5899:
5896:
5893:
5890:
5887:
5884:
5879:
5876:
5873:
5870:
5867:
5864:
5861:
5853:
5847:
5843:
5837:
5833:
5828:
5805:
5800:
5794:
5790:
5786:
5783:
5780:
5775:
5771:
5766:
5762:
5758:
5752:
5748:
5744:
5741:
5738:
5733:
5729:
5724:
5719:
5715:
5712:
5709:
5706:
5703:
5700:
5678:
5675:
5670:
5665:
5660:
5655:
5650:
5645:
5640:
5635:
5632:
5610:
5607:
5602:
5597:
5594:
5574:
5554:
5529:
5525:
5522:
5519:
5516:
5513:
5510:
5507:
5504:
5501:
5498:
5495:
5492:
5488:
5482:
5478:
5474:
5469:
5465:
5460:
5456:
5452:
5442:, the vectors
5429:
5405:
5401:
5397:
5394:
5391:
5386:
5382:
5361:
5339:
5335:
5331:
5328:
5325:
5320:
5316:
5295:
5275:
5255:
5246:Equivalently,
5244:
5243:
5229:
5205:
5201:
5178:
5174:
5170:
5167:
5164:
5159:
5155:
5143:
5129:
5105:
5101:
5074:
5070:
5066:
5063:
5060:
5055:
5051:
5024:
5021:
5017:
5011:
5007:
5003:
4998:
4994:
4989:
4985:
4980:
4975:
4972:
4969:
4965:
4942:
4939:
4934:
4930:
4926:
4923:
4920:
4915:
4911:
4890:
4887:
4882:
4878:
4874:
4871:
4868:
4863:
4859:
4838:
4828:
4816:
4795:
4775:
4753:
4750:
4747:
4744:
4741:
4738:
4735:
4732:
4728:
4706:
4686:
4666:
4646:
4626:
4623:
4620:
4617:
4614:
4594:
4591:
4588:
4585:
4582:
4579:
4576:
4556:
4534:
4531:
4528:
4509:
4504:
4501:
4500:
4499:
4485:
4482:
4479:
4457:
4454:
4451:
4431:
4428:
4425:
4422:
4419:
4413:
4410:
4404:
4401:
4398:
4395:
4392:
4389:
4386:
4365:
4361:
4355:
4352:
4346:
4343:
4321:
4318:
4315:
4312:
4309:
4306:
4300:
4297:
4266:
4263:
4260:
4257:
4254:
4251:
4248:
4224:
4221:
4218:
4196:
4193:
4190:
4170:
4167:
4164:
4161:
4158:
4155:
4152:
4149:
4146:
4143:
4139:
4116:
4113:
4110:
4089:tensor product
3998:
3995:
3969:
3966:
3963:
3941:
3938:
3935:
3932:
3929:
3918:
3917:
3906:
3903:
3899:
3895:
3892:
3889:
3886:
3883:
3869:quotient space
3851:
3848:
3845:
3823:
3820:
3815:
3811:
3807:
3802:
3798:
3794:
3791:
3769:
3766:
3761:
3757:
3753:
3748:
3744:
3740:
3737:
3724:
3723:
3708:
3705:
3702:
3699:
3696:
3693:
3690:
3687:
3684:
3682:
3680:
3677:
3674:
3671:
3668:
3665:
3662:
3661:
3658:
3655:
3652:
3649:
3646:
3643:
3640:
3637:
3634:
3632:
3630:
3627:
3624:
3621:
3618:
3615:
3612:
3611:
3608:
3605:
3600:
3596:
3592:
3589:
3586:
3583:
3580:
3575:
3571:
3567:
3564:
3561:
3558:
3555:
3553:
3551:
3546:
3542:
3538:
3533:
3529:
3525:
3522:
3519:
3516:
3515:
3512:
3509:
3506:
3503:
3498:
3494:
3490:
3487:
3484:
3481:
3478:
3473:
3469:
3465:
3462:
3459:
3457:
3455:
3452:
3449:
3444:
3440:
3436:
3431:
3427:
3423:
3420:
3419:
3369:
3366:
3363:
3360:
3357:
3333:
3330:
3327:
3324:
3321:
3301:
3298:
3295:
3292:
3289:
3264:
3261:
3258:
3236:
3233:
3230:
3210:
3207:
3204:
3201:
3198:
3167:
3164:
3161:
3117:
3114:
3082:
3060:
3036:
3033:
3030:
2999:
2996:
2993:
2973:
2970:
2967:
2947:
2944:
2941:
2938:
2935:
2932:
2929:
2926:
2923:
2920:
2916:
2891:
2887:
2862:
2858:
2837:
2834:
2831:
2828:
2825:
2822:
2802:
2799:
2796:
2793:
2790:
2787:
2761:
2758:
2755:
2752:
2746:
2742:
2736:
2732:
2724:
2720:
2716:
2713:
2709:
2701:
2697:
2693:
2690:
2686:
2682:
2679:
2677:
2675:
2670:
2665:
2659:
2655:
2647:
2643:
2639:
2636:
2632:
2626:
2621:
2616:
2611:
2605:
2601:
2593:
2589:
2585:
2582:
2578:
2572:
2567:
2564:
2562:
2560:
2557:
2554:
2551:
2550:
2530:
2527:
2524:
2504:
2501:
2498:
2467:
2463:
2459:
2454:
2450:
2427:
2401:
2398:
2395:
2373:
2370:
2367:
2364:
2361:
2358:
2355:
2339:Schauder basis
2326:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2302:
2299:
2296:
2289:
2285:
2281:
2278:
2274:
2266:
2262:
2258:
2255:
2251:
2247:
2244:
2224:
2221:
2218:
2198:
2176:
2171:
2167:
2160:
2156:
2152:
2149:
2145:
2142:
2138:
2134:
2131:
2127:
2124:
2121:
2118:
2115:
2112:
2105:
2102:
2097:
2090:
2087:
2082:
2074:
2070:
2066:
2062:
2059:
2054:
2046:
2042:
2038:
2034:
2031:
2026:
2022:
2019:
2016:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1992:
1983:, by letting:
1972:
1969:
1966:
1963:
1960:
1957:
1954:
1951:
1948:
1926:
1922:
1918:
1913:
1909:
1888:
1885:
1882:
1858:
1854:
1850:
1845:
1841:
1818:
1815:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1774:
1752:
1749:
1746:
1743:
1740:
1737:
1731:
1727:
1721:
1717:
1709:
1705:
1701:
1698:
1694:
1686:
1682:
1678:
1675:
1671:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1629:
1607:
1603:
1580:
1576:
1555:
1552:
1546:
1542:
1534:
1530:
1526:
1523:
1519:
1515:
1512:
1502:
1496:
1492:
1484:
1480:
1476:
1473:
1469:
1465:
1462:
1440:
1436:
1413:
1409:
1388:
1368:
1346:
1343:
1340:
1337:
1334:
1331:
1328:
1306:
1303:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1258:
1254:
1250:
1245:
1241:
1218:
1215:
1212:
1201:bilinear forms
1188:
1185:
1182:
1155:
1151:
1126:
1122:
1110:tensor product
1095:
1092:
1089:
1067:
1062:
1058:
1054:
1051:
1048:
1043:
1039:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
992:
989:
986:
958:
954:
950:
945:
941:
916:
913:
910:
907:
904:
884:
881:
878:
848:
844:
840:
835:
831:
803:
800:
797:
773:
769:
765:
762:
738:
734:
730:
727:
707:
704:
701:
673:
670:
667:
657:tensor product
637:
633:
608:
604:
569:
566:
538:tensor product
533:
530:
479:
476:
473:
470:
467:
440:
437:
434:
403:
400:
397:
376:are given for
361:
358:
355:
335:
332:
329:
294:
291:
288:
269:tensor product
267:is called the
256:
253:
250:
225:
222:
219:
197:
194:
191:
171:
168:
165:
162:
159:
156:
153:
147:
144:
141:
138:
135:
132:
112:
109:
106:
103:
100:
97:
94:
52:
49:
46:
36:tensor product
15:
9:
6:
4:
3:
2:
21319:
21308:
21307:Bilinear maps
21305:
21303:
21300:
21298:
21295:
21294:
21292:
21277:
21274:
21272:
21269:
21267:
21264:
21262:
21259:
21257:
21254:
21252:
21249:
21247:
21244:
21242:
21239:
21237:
21234:
21232:
21229:
21227:
21224:
21222:
21219:
21217:
21214:
21213:
21211:
21209:
21205:
21195:
21192:
21190:
21187:
21185:
21182:
21180:
21177:
21175:
21172:
21170:
21167:
21165:
21162:
21160:
21157:
21155:
21152:
21151:
21149:
21145:
21139:
21136:
21134:
21131:
21129:
21126:
21124:
21121:
21119:
21116:
21114:
21113:Metric tensor
21111:
21109:
21106:
21104:
21101:
21100:
21098:
21094:
21091:
21087:
21081:
21078:
21076:
21073:
21071:
21068:
21066:
21063:
21061:
21058:
21056:
21053:
21051:
21048:
21046:
21043:
21041:
21038:
21036:
21033:
21031:
21028:
21026:
21025:Exterior form
21023:
21021:
21018:
21016:
21013:
21011:
21008:
21006:
21003:
21001:
20998:
20996:
20993:
20991:
20988:
20987:
20985:
20979:
20972:
20969:
20967:
20964:
20962:
20959:
20957:
20954:
20952:
20949:
20947:
20944:
20942:
20939:
20937:
20934:
20932:
20929:
20927:
20924:
20922:
20919:
20918:
20916:
20914:
20910:
20904:
20901:
20899:
20898:Tensor bundle
20896:
20894:
20891:
20889:
20886:
20884:
20881:
20879:
20876:
20874:
20871:
20869:
20866:
20864:
20861:
20859:
20856:
20855:
20853:
20847:
20841:
20838:
20836:
20833:
20831:
20828:
20826:
20823:
20821:
20818:
20816:
20813:
20811:
20808:
20806:
20803:
20801:
20798:
20797:
20795:
20791:
20781:
20778:
20776:
20773:
20771:
20768:
20766:
20763:
20761:
20758:
20757:
20755:
20750:
20747:
20745:
20742:
20741:
20738:
20732:
20729:
20727:
20724:
20722:
20719:
20717:
20714:
20712:
20709:
20707:
20704:
20702:
20699:
20697:
20694:
20693:
20691:
20689:
20685:
20682:
20678:
20674:
20673:
20667:
20663:
20656:
20651:
20649:
20644:
20642:
20637:
20636:
20633:
20625:
20621:
20617:
20613:
20609:
20603:
20599:
20595:
20591:
20587:
20581:
20577:
20572:
20568:
20566:0-8218-1646-2
20562:
20558:
20554:
20550:
20546:
20543:
20539:
20535:
20531:
20527:
20521:
20517:
20513:
20509:
20505:
20501:
20495:
20491:
20487:
20483:
20479:
20477:0-387-90093-4
20473:
20469:
20465:
20461:
20457:
20451:
20447:
20442:
20438:
20434:
20430:
20426:
20422:
20420:3-540-64243-9
20416:
20412:
20408:
20404:
20403:
20390:
20384:
20380:
20373:
20365:
20361:
20357:
20351:
20347:
20343:
20339:
20332:
20321:
20314:
20305:
20300:
20293:
20285:
20279:
20268:
20264:
20257:
20250:
20242:
20240:0-387-90518-9
20236:
20232:
20225:
20218:
20212:
20197:
20193:
20189:
20183:
20176:
20175:inner product
20172:
20171:contravariant
20166:
20159:
20155:
20151:
20146:
20138:
20132:
20128:
20121:
20114:
20109:
20107:
20099:
20094:
20092:
20087:
20076:
20073:
20070:
20067:
20064:
20061:
20058:
20055:
20053:
20050:
20047:
20044:
20043:
20037:
20035:
20031:
20027:
20023:
20018:
20012:
19991:
19982:(for example
19977:
19972:A ○.× B ○.× C
19966:(for example
19961:
19957:
19943:
19941:
19936:
19920:
19917:
19912:
19908:
19904:
19899:
19896:
19893:
19889:
19885:
19882:
19879:
19876:
19873:
19868:
19865:
19862:
19858:
19854:
19849:
19845:
19841:
19838:
19823:
19817:
19813:
19810:
19807:
19804:
19801:
19794:
19791:
19788:
19783:
19779:
19769:
19756:
19744:
19740:
19736:
19728:
19724:
19720:
19715:
19711:
19704:
19699:
19695:
19691:
19686:
19682:
19678:
19673:
19669:
19665:
19660:
19656:
19638:
19635:
19632:
19629:
19626:
19623:
19620:
19612:
19607:
19605:
19602:
19598:differential
19594:
19589:
19584:
19565:
19560:
19544:
19527:
19524:
19521:
19518:
19515:
19491:
19487:
19483:
19478:
19474:
19470:
19467:
19462:
19458:
19454:
19449:
19445:
19420:
19416:
19412:
19407:
19403:
19380:
19376:
19372:
19367:
19363:
19355:The image of
19353:
19340:
19328:
19324:
19320:
19312:
19308:
19304:
19299:
19295:
19288:
19283:
19279:
19275:
19270:
19266:
19262:
19257:
19253:
19249:
19244:
19240:
19222:
19219:
19216:
19213:
19210:
19207:
19204:
19195:
19189:
19176:
19170:
19167:
19164:
19161:
19158:
19155:
19152:
19139:
19136:
19133:
19130:
19127:
19124:
19121:
19101:
19098:
19095:
19086:
19081:
19076:
19074:
19070:
19066:
19062:
19058:
19054:
19050:
19040:
19038:
19028:
19026:
19022:
19018:
19014:
19010:
19004:
18993:
18982:
18976:
18965:
18955:
18953:
18949:
18945:
18940:
18927:
18919:
18916:
18913:
18909:
18905:
18902:
18899:
18894:
18891:
18888:
18884:
18877:
18869:
18865:
18861:
18858:
18855:
18850:
18846:
18839:
18836:
18828:
18825:
18822:
18818:
18814:
18811:
18808:
18803:
18799:
18789:
18786:
18783:
18760:
18740:
18715:
18711:
18707:
18704:
18701:
18696:
18692:
18685:
18660:
18656:
18652:
18649:
18646:
18641:
18637:
18630:
18623:
18612:
18602:
18600:
18596:
18590:
18580:
18578:
18572:
18562:
18560:
18555:
18553:
18548:
18544:
18540:
18536:
18531:
18527:
18512:
18510:
18495:
18492:
18489:
18481:
18465:
18462:
18457:
18438:
18434:
18428:
18410:
18406:
18387:
18383:
18373:
18369:
18363:
18356:
18352:
18344:
18340:
18333:
18304:
18281:
18275:
18271:
18267:
18264:
18261:
18256:
18252:
18247:
18243:
18216:
18213:
18210:
18188:
18184:
18176:
18160:
18140:
18131:
18118:
18115:
18112:
18109:
18106:
18103:
18100:
18084:
18080:
18075:
18071:
18064:
18060:
18055:
18047:
18043:
18038:
18030:
18026:
18022:
18017:
18013:
18007:
18003:
17999:
17995:
17989:
17984:
17981:
17976:
17972:
17967:
17963:
17958:
17953:
17950:
17945:
17941:
17936:
17930:
17925:
17922:
17917:
17913:
17908:
17904:
17896:
17892:
17888:
17885:
17882:
17877:
17873:
17864:
17860:
17837:
17834:
17831:
17816:
17813:
17810:
17781:
17777:
17768:
17764:
17756:
17738:
17735:
17725:
17721:
17714:
17711:
17703:
17687:
17660:
17656:
17652:
17647:
17643:
17637:
17633:
17628:
17617:with entries
17604:
17601:
17598:
17595:
17592:
17589:
17586:
17566:
17539:
17535:
17531:
17526:
17522:
17516:
17512:
17507:
17500:
17497:
17489:
17473:
17453:
17429:
17426:
17423:
17394:
17374:
17366:
17348:
17326:
17318:
17314:
17298:
17290:
17274:
17252:
17242:
17224:
17221:
17218:
17203:
17200:
17197:
17187:
17184:
17160:
17156:
17147:
17143:
17132:
17131:vector spaces
17128:
17112:
17105:
17089:
17082:
17072:
17051:
17045:
17042:
17038:
17026:
17006:
17000:
16996:
16989:
16983:
16980:
16977:
16972:
16968:
16964:
16955:
16950:
16945:
16941:
16935:
16929:
16923:
16903:
16897:
16893:
16886:
16880:
16877:
16874:
16869:
16865:
16861:
16852:
16847:
16842:
16835:
16831:
16827:
16823:
16818:
16817:Galois theory
16814:
16809:
16803:
16797:
16791:
16778:
16772:
16769:
16766:
16760:
16757:
16751:
16745:
16740:
16736:
16729:
16723:
16716:For example:
16703:
16695:
16691:
16687:
16682:
16678:
16671:
16663:
16659:
16655:
16650:
16646:
16639:
16631:
16627:
16623:
16618:
16614:
16607:
16599:
16595:
16591:
16586:
16582:
16569:
16552:
16547:
16543:
16539:
16531:
16528:
16521:
16515:
16509:
16503:
16496:
16486:
16484:
16480:
16475:
16472:
16468:
16464:
16461:
16457:
16450:
16447:
16443:
16437:
16433:
16429:
16423:
16406:
16401:
16397:
16391:
16387:
16380:
16375:
16371:
16365:
16361:
16336:
16332:
16323:
16319:
16307:
16302:
16284:
16279:
16275:
16271:
16260:
16239:
16235:
16212:
16207:
16204:
16200:
16177:
16173:
16163:
16146:
16143:
16140:
16118:
16114:
16105:
16101:
16078:
16073:
16070:
16067:
16063:
16059:
16054:
16050:
16038:
16024:
16018:
16014:
16005:
16001:
15996:
15992:
15989:
15986:
15983:
15978:
15974:
15970:
15962:
15946:
15943:
15940:
15935:
15932:
15928:
15923:
15920:
15917:
15912:
15908:
15902:
15899:
15895:
15889:
15886:
15883:
15879:
15858:
15855:
15852:
15849:
15846:
15843:
15838:
15834:
15824:
15818:
15813:
15808:
15795:
15792:
15789:
15785:
15781:
15778:
15770:
15766:
15749:
15745:
15738:is given by:
15736:
15733:
15729:
15723:
15717:
15714:
15710:
15691:
15688:
15685:
15674:
15668:
15651:
15648:
15645:
15628:
15624:
15608:
15603:
15599:
15595:
15573:
15570:
15567:
15564:
15561:
15539:
15536:
15533:
15530:
15527:
15519:
15515:
15511:
15507:
15503:
15499:
15496:
15492:
15476:
15471:
15467:
15463:
15455:
15451:
15447:
15443:
15439:
15435:
15431:
15411:
15408:
15402:
15399:
15396:
15393:
15387:
15384:
15381:
15368:
15352:
15347:
15343:
15339:
15331:
15327:
15323:
15319:
15302:
15299:
15293:
15290:
15284:
15278:
15275:
15272:
15266:
15256:
15240:
15235:
15231:
15227:
15219:
15215:
15211:
15207:
15206:
15205:
15191:
15186:
15182:
15178:
15169:
15153:
15150:
15147:
15125:
15122:
15119:
15096:
15093:
15090:
15084:
15081:
15075:
15072:
15069:
15058:
15036:
15030:
15027:
15024:
15021:
15013:
15002:
14985:
14979:
14976:
14973:
14970:
14964:
14961:
14959:
14951:
14948:
14945:
14942:
14936:
14929:
14921:
14917:
14913:
14910:
14904:
14901:
14893:
14889:
14885:
14882:
14876:
14873:
14871:
14861:
14857:
14853:
14848:
14844:
14840:
14837:
14831:
14824:
14818:
14815:
14810:
14806:
14799:
14796:
14790:
14787:
14782:
14778:
14771:
14768:
14766:
14758:
14755:
14750:
14746:
14742:
14737:
14733:
14726:
14702:
14696:
14693:
14690:
14687:
14684:
14676:
14671:
14654:
14648:
14645:
14642:
14639:
14633:
14627:
14624:
14621:
14618:
14613:
14605:
14597:
14593:
14589:
14584:
14580:
14576:
14573:
14567:
14559:
14555:
14551:
14548:
14542:
14534:
14530:
14526:
14523:
14518:
14510:
14504:
14501:
14496:
14492:
14488:
14483:
14479:
14472:
14466:
14463:
14458:
14454:
14447:
14441:
14438:
14433:
14429:
14423:
14415:
14412:
14409:
14406:
14398:
14395:
14392:
14387:
14383:
14379:
14374:
14370:
14366:
14363:
14357:
14354:
14351:
14346:
14342:
14338:
14333:
14329:
14325:
14322:
14291:
14288:
14285:
14279:
14259:
14256:
14253:
14245:
14226:
14223:
14220:
14214:
14194:
14190:
14183:
14180:
14177:
14171:
14168:
14165:
14160:
14156:
14152:
14144:
14140:
14136:
14132:
14128:
14124:
14114:
14113:for details.
14112:
14096:
14074:
14071:
14068:
14065:
14062:
14040:
14035:
14031:
14027:
14018:
13999:
13996:
13993:
13971:
13949:
13946:
13943:
13934:
13933:abelian group
13929:
13923:
13903:
13900:
13897:
13894:
13888:
13885:
13883:
13875:
13872:
13869:
13866:
13860:
13849:
13846:
13842:
13839:
13833:
13830:
13824:
13821:
13818:
13812:
13809:
13807:
13798:
13795:
13791:
13788:
13785:
13782:
13776:
13766:
13763:
13759:
13756:
13749:
13746:
13740:
13737:
13734:
13728:
13725:
13723:
13715:
13712:
13708:
13705:
13701:
13698:
13692:
13680:
13664:
13661:
13658:
13649:
13646:
13643:
13620:
13615:
13611:
13607:
13601:
13598:
13595:
13592:
13589:
13580:
13578:
13577:abelian group
13573:
13567:
13547:
13544:
13541:
13538:
13532:
13526:
13523:
13520:
13517:
13505:
13499:
13493:
13487:
13482:
13477:
13475:
13470:
13464:
13459:
13456:
13434:
13431:
13428:
13422:
13402:
13399:
13395:
13388:
13385:
13382:
13376:
13373:
13370:
13365:
13361:
13357:
13348:
13344:
13341:
13340:
13334:
13328:
13324:
13318:
13308:
13306:
13290:
13281:
13278:
13275:
13258:
13255:
13238:
13232:
13229:
13226:
13223:
13220:
13198:
13193:
13187:
13181:
13175:
13172:
13166:
13162:are bases of
13144:
13140:
13133:
13125:
13121:
13097:
13094:
13091:
13066:
13062:
13058:
13053:
13049:
13025:
13019:
13013:
13010:
13004:
12998:
12995:
12992:
12986:
12983:
12980:
12974:
12971:
12962:
12942:
12934:
12930:
12923:
12920:
12915:
12910:
12906:
12900:
12896:
12889:
12882:
12879:
12874:
12870:
12860:
12857:
12854:
12834:
12824:
12803:
12798:
12794:
12765:
12761:
12748:
12735:
12732:
12726:
12720:
12717:
12711:
12702:
12699:
12696:
12671:
12649:
12646:
12641:
12637:
12633:
12630:
12627:
12624:
12604:
12598:
12595:
12592:
12575:
12572:
12569:
12564:
12560:
12549:
12545:
12538:
12532:
12526:
12520:
12514:
12509:
12506:, denote the
12504:
12498:
12492:
12481:
12462:
12436:
12431:
12410:
12371:
12363:
12358:
12354:
12330:
12307:
12301:
12288:
12285:
12279:
12276:
12273:
12267:
12261:
12252:
12249:
12246:
12220:
12197:
12189:
12184:
12180:
12170:
12157:
12148:
12140:
12136:
12132:
12129:
12123:
12117:
12114:
12108:
12102:
12075:
12071:
12067:
12064:
12053:
12048:
12031:
12026:
12022:
12018:
12003:
11998:
11994:
11973:
11967:
11959:
11955:
11951:
11948:
11945:
11942:
11939:
11933:
11927:
11924:
11918:
11915:
11912:
11906:
11881:
11864:
11861:
11837:
11834:
11831:
11828:
11825:
11800:
11779:
11760:
11752:
11747:
11743:
11728:
11725:
11723:
11705:
11700:
11696:
11673:
11649:
11645:
11641:
11638:
11635:
11630:
11626:
11596:
11591:
11587:
11583:
11578:
11574:
11570:
11565:
11561:
11554:
11545:
11541:
11537:
11534:
11528:
11522:
11512:
11506:
11492:
11483:
11467:
11463:
11442:
11422:
11416:
11408:
11405:
11402:
11397:
11394:
11391:
11387:
11377:
11369:
11364:
11360:
11351:
11345:
11341:
11319:
11316:
11313:
11299:
11286:
11280:
11274:
11268:
11265:
11262:
11242:
11234:
11230:
11226:
11223:
11216:
11202:
11200:
11197:, called the
11196:
11191:
11178:
11173:
11164:
11160:
11154:
11150:
11146:
11141:
11132:
11129:
11121:
11118:
11115:
11090:
11086:
11080:
11071:
11067:
11063:
11058:
11049:
11040:
11035:
11031:
11028:
11025:
11021:
10995:
10991:
10965:
10962:
10959:
10947:
10926:
10921:
10917:
10901:
10895:
10882:
10875:
10872:
10869:
10865:
10861:
10856:
10853:
10850:
10846:
10840:
10837:
10834:
10830:
10824:
10821:
10818:
10814:
10809:
10801:
10797:
10793:
10788:
10784:
10778:
10774:
10769:
10765:
10758:
10755:
10752:
10748:
10744:
10739:
10735:
10729:
10725:
10716:
10713:
10710:
10683:
10678:
10674:
10670:
10667:
10643:
10638:
10634:
10630:
10627:
10618:
10612:
10607:
10602:
10596:
10591:
10587:
10583:
10564:
10556:
10551:
10547:
10524:
10520:
10511:
10506:
10500:
10494:
10481:
10476:
10472:
10468:
10464:
10460:
10456:
10452:
10447:
10443:
10439:
10432:
10428:
10424:
10417:
10409:
10405:
10401:
10396:
10392:
10366:
10362:
10352:
10333:
10329:
10319:
10313:
10300:
10294:
10285:
10282:
10278:
10275:
10269:
10266:
10262:
10259:
10255:
10245:
10236:
10233:
10226:
10223:
10218:
10212:
10208:
10201:
10193:
10188:
10184:
10173:
10170:
10167:
10161:
10155:
10151:
10147:
10129:
10125:
10104:
10099:
10096:
10091:
10086:
10082:
10078:
10073:
10068:
10065:
10061:
10057:
10052:
10046:
10040:
10036:
10032:
10029:
10026:
10021:
10017:
10009:
10004:
9998:
9994:
9991:
9988:
9985:
9982:
9975:
9969:
9961:
9956:
9952:
9942:
9937:
9919:
9916:
9913:
9901:
9895:
9888:
9878:
9876:
9871:
9855:
9852:
9849:
9846:
9843:
9840:
9837:
9834:
9831:
9828:
9825:
9822:
9819:
9809:
9805:
9799:
9796:
9783:
9778:
9770:
9767:
9764:
9760:
9754:
9751:
9748:
9744:
9736:
9733:
9730:
9726:
9720:
9717:
9714:
9710:
9702:
9699:
9696:
9692:
9686:
9683:
9680:
9676:
9668:
9665:
9662:
9658:
9652:
9649:
9646:
9642:
9632:
9629:
9626:
9622:
9616:
9613:
9610:
9606:
9598:
9595:
9592:
9588:
9582:
9579:
9576:
9572:
9564:
9561:
9558:
9554:
9548:
9545:
9542:
9538:
9530:
9527:
9524:
9520:
9514:
9511:
9508:
9504:
9494:
9491:
9488:
9484:
9478:
9475:
9472:
9468:
9460:
9457:
9454:
9450:
9444:
9441:
9438:
9434:
9426:
9423:
9420:
9416:
9410:
9407:
9404:
9400:
9392:
9389:
9386:
9382:
9376:
9373:
9370:
9366:
9356:
9353:
9350:
9346:
9340:
9337:
9334:
9330:
9322:
9319:
9316:
9312:
9306:
9303:
9300:
9296:
9288:
9285:
9282:
9278:
9272:
9269:
9266:
9262:
9254:
9251:
9248:
9244:
9238:
9235:
9232:
9228:
9221:
9216:
9211:
9203:
9195:
9192:
9189:
9185:
9177:
9174:
9171:
9167:
9157:
9154:
9151:
9147:
9139:
9136:
9133:
9129:
9122:
9115:
9112:
9109:
9105:
9097:
9089:
9086:
9083:
9079:
9071:
9068:
9065:
9061:
9051:
9048:
9045:
9041:
9033:
9030:
9027:
9023:
9016:
9009:
9006:
9003:
8999:
8989:
8981:
8978:
8975:
8971:
8963:
8960:
8957:
8953:
8943:
8940:
8937:
8933:
8925:
8922:
8919:
8915:
8908:
8901:
8898:
8895:
8891:
8883:
8875:
8872:
8869:
8865:
8857:
8854:
8851:
8847:
8837:
8834:
8831:
8827:
8819:
8816:
8813:
8809:
8802:
8795:
8792:
8789:
8785:
8778:
8773:
8768:
8760:
8757:
8754:
8750:
8742:
8739:
8736:
8732:
8722:
8719:
8716:
8712:
8704:
8701:
8698:
8694:
8687:
8682:
8677:
8669:
8666:
8663:
8659:
8651:
8648:
8645:
8641:
8631:
8628:
8625:
8621:
8613:
8610:
8607:
8603:
8596:
8574:
8569:
8561:
8558:
8555:
8551:
8543:
8540:
8537:
8533:
8523:
8520:
8517:
8513:
8505:
8502:
8499:
8495:
8488:
8483:
8480:
8476:
8471:
8463:
8460:
8457:
8453:
8445:
8442:
8439:
8435:
8425:
8422:
8419:
8415:
8407:
8404:
8401:
8397:
8390:
8385:
8382:
8373:
8367:
8361:
8355:
8351:
8347:
8342:
8326:
8323:
8320:
8300:
8297:
8294:
8286:
8281:
8275:
8269:
8267:
8263:
8259:
8255:
8254:exact functor
8251:
8247:
8234:
8232:
8228:
8224:
8205:
8199:
8196:
8193:
8187:
8181:
8178:
8175:
8169:
8163:
8160:
8157:
8151:
8145:
8142:
8139:
8133:
8130:
8127:
8124:
8117:
8116:
8115:
8098:
8092:
8086:
8083:
8077:
8071:
8068:
8062:
8059:
8056:
8047:
8044:
8041:
8031:
8030:
8029:
8012:
8009:
8006:
8000:
7997:
7994:
7991:
7988:
7985:
7982:
7975:
7974:
7973:
7957:
7951:
7948:
7945:
7923:
7917:
7914:
7911:
7902:
7888:
7885:
7882:
7859:
7856:
7853:
7847:
7841:
7838:
7832:
7829:
7826:
7817:
7814:
7811:
7801:
7800:
7799:
7782:
7779:
7776:
7770:
7767:
7764:
7761:
7758:
7755:
7752:
7745:
7744:
7743:
7742:
7720:
7714:
7711:
7708:
7695:
7685:
7667:
7664:
7660:
7651:
7648:
7644:
7613:
7607:
7603:
7599:
7596:
7593:
7585:
7579:
7575:
7566:
7562:
7558:
7555:
7552:
7547:
7543:
7535:
7534:
7533:
7528:of the first
7524:
7498:
7495:
7491:
7482:
7478:
7470:
7454:
7451:
7448:
7428:
7425:
7422:
7416:
7413:
7410:
7401:
7385:
7382:
7379:
7376:
7373:
7370:
7367:
7343:
7340:
7337:
7326:
7310:
7307:
7304:
7282:
7279:
7276:
7253:
7250:
7247:
7244:
7241:
7238:
7235:
7232:
7225:
7224:
7223:
7221:
7205:
7185:
7171:
7168:
7149:
7146:
7143:
7137:
7134:
7112:
7109:
7103:
7100:
7097:
7071:
7065:
7062:
7059:
7053:
7050:
7047:
7044:
7041:
7035:
7032:
7029:
7019:
7018:
7017:
7001:
6998:
6995:
6992:
6989:
6979:
6972:Associativity
6969:
6966:
6960:
6943:
6940:
6937:
6928:
6925:
6919:
6902:
6899:
6896:
6888:
6883:
6877:
6861:
6845:
6823:
6799:
6796:
6793:
6777:
6774:
6771:
6759:
6746:
6743:
6740:
6710:
6707:
6704:
6676:
6672:
6669:
6666:
6663:
6660:
6657:
6654:
6651:
6648:
6645:
6642:
6638:
6634:
6631:
6628:
6625:
6603:
6593:
6590:
6568:
6558:
6555:
6530:
6524:
6518:
6512:
6503:
6500:
6497:
6470:
6467:
6464:
6454:
6451:
6448:
6445:
6421:
6411:
6408:
6384:
6374:
6371:
6351:
6331:
6306:
6300:
6297:
6291:
6269:
6266:
6243:
6237:
6234:
6228:
6222:
6216:
6196:
6193:
6190:
6170:
6148:
6132:
6116:
6113:
6110:
6107:
6101:
6098:
6095:
6067:
6064:
6061:
6055:
6051:
6047:
6044:
6041:
6020:
5999:
5975:
5965:
5962:
5938:
5928:
5925:
5900:
5897:
5894:
5891:
5888:
5885:
5882:
5877:
5874:
5871:
5868:
5865:
5862:
5859:
5851:
5845:
5841:
5835:
5831:
5826:
5803:
5798:
5792:
5788:
5784:
5781:
5778:
5773:
5769:
5764:
5760:
5756:
5750:
5746:
5742:
5739:
5736:
5731:
5727:
5722:
5717:
5713:
5707:
5704:
5701:
5676:
5673:
5658:
5648:
5643:
5633:
5630:
5608:
5605:
5595:
5592:
5572:
5552:
5542:
5527:
5523:
5520:
5517:
5514:
5511:
5508:
5505:
5502:
5499:
5496:
5493:
5490:
5486:
5480:
5476:
5472:
5467:
5463:
5458:
5454:
5450:
5427:
5403:
5399:
5395:
5392:
5389:
5384:
5380:
5359:
5337:
5333:
5329:
5326:
5323:
5318:
5314:
5293:
5273:
5253:
5227:
5203:
5199:
5176:
5172:
5168:
5165:
5162:
5157:
5153:
5144:
5127:
5103:
5099:
5090:
5072:
5068:
5064:
5061:
5058:
5053:
5049:
5040:
5039:
5038:
5022:
5019:
5015:
5009:
5005:
5001:
4996:
4992:
4987:
4983:
4978:
4973:
4970:
4967:
4963:
4940:
4937:
4932:
4928:
4924:
4921:
4918:
4913:
4909:
4888:
4885:
4880:
4876:
4872:
4869:
4866:
4861:
4857:
4836:
4814:
4807:
4793:
4773:
4751:
4748:
4742:
4739:
4736:
4730:
4726:
4704:
4697:spans all of
4684:
4664:
4644:
4621:
4618:
4615:
4592:
4586:
4583:
4580:
4577:
4574:
4554:
4532:
4529:
4526:
4508:
4483:
4480:
4477:
4455:
4452:
4449:
4426:
4423:
4420:
4408:
4402:
4396:
4393:
4390:
4384:
4363:
4359:
4350:
4344:
4341:
4319:
4313:
4310:
4307:
4304:
4295:
4282:
4279:, there is a
4264:
4258:
4255:
4252:
4249:
4246:
4222:
4219:
4216:
4194:
4191:
4188:
4168:
4165:
4162:
4153:
4150:
4147:
4141:
4137:
4114:
4111:
4108:
4090:
4086:
4085:
4084:
4082:
4079:
4075:
4070:
4066:
4064:
4060:
4051:
4044:
4034:
4029:
4021:
4009:
4003:
3994:
3992:
3988:
3983:
3967:
3964:
3961:
3936:
3933:
3930:
3904:
3901:
3897:
3893:
3890:
3887:
3884:
3881:
3874:
3873:
3872:
3870:
3865:
3849:
3846:
3843:
3821:
3818:
3813:
3809:
3805:
3800:
3796:
3792:
3789:
3767:
3764:
3759:
3755:
3751:
3746:
3742:
3738:
3735:
3706:
3700:
3697:
3694:
3688:
3685:
3683:
3675:
3672:
3669:
3666:
3656:
3650:
3647:
3644:
3638:
3635:
3633:
3625:
3622:
3619:
3616:
3606:
3598:
3594:
3590:
3587:
3581:
3573:
3569:
3565:
3562:
3556:
3554:
3544:
3540:
3536:
3531:
3527:
3523:
3520:
3510:
3504:
3501:
3496:
3492:
3485:
3479:
3476:
3471:
3467:
3460:
3458:
3450:
3447:
3442:
3438:
3434:
3429:
3425:
3410:
3409:
3408:
3406:
3394:
3385:
3364:
3361:
3358:
3328:
3325:
3322:
3299:
3293:
3290:
3287:
3280:
3262:
3259:
3256:
3234:
3231:
3228:
3205:
3202:
3199:
3189:
3181:
3165:
3162:
3159:
3152:
3149:that has the
3143:
3141:
3134:
3133:vector spaces
3121:
3113:
3111:
3107:
3102:
3096:
3080:
3058:
3050:
3049:outer product
3034:
3031:
3028:
3020:
3015:
3013:
2997:
2994:
2991:
2971:
2968:
2965:
2945:
2942:
2939:
2930:
2927:
2924:
2918:
2914:
2889:
2885:
2860:
2856:
2832:
2829:
2826:
2820:
2797:
2794:
2791:
2785:
2776:
2759:
2756:
2753:
2750:
2744:
2740:
2734:
2730:
2722:
2718:
2714:
2711:
2707:
2699:
2695:
2691:
2688:
2684:
2680:
2678:
2663:
2657:
2653:
2645:
2641:
2637:
2634:
2630:
2619:
2609:
2603:
2599:
2591:
2587:
2583:
2580:
2576:
2565:
2563:
2558:
2555:
2552:
2528:
2525:
2522:
2502:
2499:
2496:
2488:
2483:
2465:
2461:
2457:
2452:
2448:
2425:
2417:
2399:
2396:
2393:
2368:
2365:
2362:
2359:
2356:
2340:
2321:
2318:
2315:
2306:
2303:
2300:
2294:
2287:
2283:
2279:
2276:
2272:
2264:
2260:
2256:
2253:
2249:
2245:
2242:
2222:
2219:
2216:
2196:
2187:
2174:
2169:
2165:
2158:
2154:
2150:
2143:
2140:
2136:
2132:
2129:
2119:
2116:
2113:
2103:
2100:
2095:
2088:
2085:
2080:
2072:
2068:
2064:
2060:
2057:
2052:
2044:
2040:
2036:
2032:
2029:
2024:
2020:
2014:
2011:
2008:
1999:
1996:
1993:
1970:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1924:
1920:
1916:
1911:
1907:
1886:
1883:
1880:
1856:
1852:
1848:
1843:
1839:
1816:
1813:
1810:
1807:
1801:
1798:
1795:
1772:
1763:
1747:
1744:
1741:
1735:
1729:
1725:
1719:
1715:
1707:
1703:
1699:
1696:
1692:
1684:
1680:
1676:
1673:
1669:
1665:
1659:
1656:
1653:
1647:
1627:
1605:
1601:
1578:
1574:
1553:
1550:
1544:
1540:
1532:
1528:
1524:
1521:
1517:
1513:
1510:
1500:
1494:
1490:
1482:
1478:
1474:
1471:
1467:
1463:
1460:
1438:
1434:
1411:
1407:
1399:in the bases
1386:
1366:
1344:
1338:
1335:
1332:
1329:
1326:
1304:
1301:
1298:
1295:
1289:
1286:
1283:
1256:
1252:
1248:
1243:
1239:
1216:
1213:
1210:
1202:
1186:
1183:
1180:
1171:
1153:
1149:
1124:
1120:
1112:of the bases
1111:
1093:
1090:
1087:
1060:
1056:
1052:
1049:
1046:
1041:
1037:
1033:
1030:
1027:
1024:
1021:
1018:
1006:
990:
987:
984:
956:
952:
948:
943:
939:
911:
908:
905:
882:
879:
876:
868:
846:
842:
838:
833:
829:
821:
817:
801:
798:
795:
771:
767:
763:
760:
736:
732:
728:
725:
705:
702:
699:
671:
668:
665:
658:
653:
635:
631:
606:
602:
594:
587:
583:
582:vector spaces
565:
562:
558:
554:
549:
547:
543:
539:
529:
527:
523:
520:
516:
512:
509:, which is a
508:
507:metric tensor
504:
500:
495:
493:
477:
471:
468:
465:
458:
438:
435:
432:
423:
401:
398:
395:
388:, a basis of
386:
380:
375:
359:
356:
353:
333:
330:
327:
320:
316:
312:
308:
292:
289:
286:
270:
254:
251:
248:
239:
223:
220:
217:
195:
192:
189:
169:
166:
163:
160:
157:
154:
151:
145:
139:
136:
133:
110:
107:
104:
98:
95:
92:
85:
81:
76:
70:
66:
65:vector spaces
50:
47:
44:
37:
33:
26:
22:
21276:Hermann Weyl
21080:Vector space
21065:Pseudotensor
21030:Fiber bundle
20983:abstractions
20965:
20878:Mixed tensor
20863:Tensor field
20670:
20597:
20575:
20556:
20553:Birkhoff, G.
20549:Mac Lane, S.
20511:
20492:. Springer.
20489:
20470:. Springer.
20467:
20464:Halmos, Paul
20445:
20410:
20378:
20372:
20337:
20331:
20313:
20292:
20262:
20249:
20233:. Springer.
20230:
20224:
20211:
20200:. Retrieved
20191:
20182:
20165:
20157:
20153:
20145:
20126:
20120:
20028:such as the
20019:
19992:
19954:
19937:
19770:
19608:
19600:
19592:
19582:
19542:
19354:
19193:
19190:
19084:
19077:
19075:in general.
19069:Weyl algebra
19046:
19034:
19006:
18941:
18619:
18592:
18574:
18556:
18533:
18132:
17387:consists of
17364:
17317:eigenvectors
17313:well-defined
17078:
17024:
16953:
16943:
16939:
16933:
16927:
16921:
16850:
16840:
16833:
16829:
16825:
16821:
16807:
16801:
16795:
16792:
16567:
16526:
16519:
16513:
16507:
16501:
16498:
16479:Tor functors
16473:
16470:
16466:
16462:
16459:
16455:
16448:
16445:
16441:
16435:
16431:
16427:
16421:
16305:
16258:
16161:
16039:
15822:
15816:
15812:presentation
15809:
15734:
15731:
15727:
15721:
15715:
15712:
15708:
15672:
15666:
15637:
15626:
15622:
15517:
15513:
15509:
15505:
15501:
15494:
15490:
15453:
15449:
15445:
15441:
15437:
15433:
15366:
15329:
15325:
15321:
15254:
15217:
15213:
15209:
15170:
15003:
14674:
14672:
14142:
14138:
14134:
14130:
14129:-module and
14126:
14122:
14120:
14111:main article
14016:
13927:
13924:
13581:
13571:
13565:
13503:
13497:
13495:-module and
13491:
13485:
13478:
13468:
13462:
13454:
13346:
13337:
13332:
13326:
13320:
13196:
13191:
13185:
13179:
13176:
13170:
13164:
12963:
12749:
12547:
12543:
12536:
12530:
12524:
12518:
12512:
12502:
12496:
12490:
12487:
12434:
12305:
12302:
12171:
12051:
11734:
11726:
11510:
11484:
11343:
11339:
11300:
11214:
11208:
11192:
10945:
10899:
10896:
10616:
10610:
10600:
10594:
10504:
10498:
10495:
10350:
10317:
10314:
10171:
10165:
10159:
10153:
9940:
9899:
9893:
9890:
9872:
9797:
8371:
8365:
8359:
8353:
8349:
8345:
8279:
8273:
8270:
8235:
8221:In terms of
8220:
8113:
8027:
7903:
7874:
7797:
7740:
7698:
7633:
7521:. For every
7475:braiding map
7472:
7469:automorphism
7402:
7327:
7268:
7220:commutative
7177:
7169:
7086:
6975:
6964:
6958:
6929:
6923:
6917:
6881:
6875:
6872:
6133:
5544:
5245:
4766:), and also
4510:
4506:
4334:, such that
4280:
4088:
4077:
4074:bilinear map
4071:
4067:
4056:
4049:
4042:
4032:
4019:
4007:
3984:
3919:
3866:
3725:
3386:
3144:
3122:
3119:
3097:
3016:
2777:
2486:
2484:
2188:
1764:
1172:
1109:
1007:
656:
654:
571:
550:
537:
535:
515:vector field
513:(so like an
511:tensor field
496:
424:
384:
378:
314:
310:
268:
240:
84:bilinear map
74:
68:
35:
29:
21216:Élie Cartan
21164:Spin tensor
21138:Weyl tensor
21096:Mathematics
21060:Multivector
20851:definitions
20749:Engineering
20688:Mathematics
20508:Lang, Serge
20113:Trèves 2006
20098:Trèves 2006
19988:a */ b */ c
18577:direct sums
17446:, provided
17289:nonsingular
17127:linear maps
16819:: if, say,
16165:th copy of
15629:)-bimodule.
15365:is a right
14125:be a right
13633:defined by
13339:commutative
11778:Lie algebra
10150:linear maps
9808:matrix rank
9804:tensor rank
7523:permutation
6978:associative
6282:is the map
6209:is the map
4283:linear map
4063:isomorphism
4028:commutative
3384:otherwise.
2541:as before:
1899:defined on
975:is denoted
546:isomorphism
32:mathematics
21291:Categories
21045:Linear map
20913:Operations
20542:0984.00001
20499:0387905189
20400:References
20304:1505.05729
20202:2017-01-26
19071:, and the
18979:See also:
18620:Given two
18203:of degree
17407:points in
17125:represent
16919:where now
15253:is a left
15049:such that
14133:be a left
14053:satisfies
13501:is a left-
13415:where now
12522:, and the
12508:dual space
12236:given by:
11722:dual basis
10943:, and let
10586:components
10510:dual basis
9885:See also:
8250:surjective
8114:One has:
7269:that maps
7087:that maps
6864:Properties
4953:such that
4717:(that is,
4442:for every
4377:(that is,
4078:separately
4030:(that is,
3405:spanned by
568:From bases
519:space-time
457:linear map
21184:EM tensor
21020:Dimension
20971:Transpose
20616:853623322
20596:(2006) .
19921:…
19918:⊗
19905:⊗
19886:⊗
19883:⋯
19880:−
19877:⋯
19874:⊗
19855:⊗
19842:⊗
19839:⋯
19818:⏟
19811:⊗
19808:⋯
19805:⊗
19789:
19737:∈
19705:∣
19692:⊗
19679:−
19666:⊗
19636:⊗
19624:⊙
19557:Λ
19525:⊗
19522:⋯
19519:⊗
19484:∧
19471:−
19455:∧
19413:∧
19373:⊗
19321:∈
19289:∣
19276:⊗
19250:⊗
19220:⊗
19208:∧
19168:∈
19162:∣
19156:⊗
19137:⊗
19125:∧
19099:∧
19053:quotients
18903:…
18859:…
18812:…
18787:⊗
18705:…
18650:…
18493:×
18463:≤
18435:ψ
18429:⋯
18407:ψ
18384:ψ
18364:⋯
18265:…
18214:−
18185:ψ
18161:ψ
18113:…
18096:for
18072:⋯
18023:⋯
17968:∑
17964:⋯
17937:∑
17909:∑
17886:…
17861:ψ
17835:−
17822:→
17814:−
17774:→
17736:⊗
17715:∈
17653:⋯
17602:×
17599:⋯
17596:×
17590:×
17532:⋯
17427:−
17349:ψ
17299:ψ
17222:−
17209:→
17201:−
17185:ψ
17153:→
17046:
16981:≅
16969:⊗
16878:≅
16866:⊗
16758:≅
16737:⊗
16688:⋅
16672:⊗
16656:⋅
16624:⊗
16608:⋅
16592:⊗
16544:⊗
16530:-algebras
16454:0 :
16398:⊗
16384:→
16372:⊗
16329:→
16309:-modules
16276:⊗
16144:∈
16111:→
16071:∈
16064:⊕
16011:→
15993:
15975:⊗
15941:∈
15887:∈
15880:∑
15856:∈
15844:∈
15750:⊗
15689:⊗
15649:⊗
15600:⊗
15468:⊗
15403:⊗
15385:⊗
15344:⊗
15300:⊗
15276:⊗
15232:⊗
15183:⊗
15151:∈
15123:∈
15073:⊗
15062:¯
15034:→
15028:⊗
15017:¯
14700:→
14694:×
14634:−
14568:−
14473:−
14410:∈
14393:∈
14361:∀
14352:∈
14320:∀
14289:×
14257:×
14224:×
14181:×
14157:⊗
14097:φ
14075:φ
14072:∘
14063:ψ
14032:⊗
13997:×
13972:ψ
13947:×
13889:φ
13861:φ
13834:φ
13813:φ
13777:φ
13750:φ
13729:φ
13693:φ
13662:⊗
13656:↦
13612:⊗
13605:→
13599:×
13590:φ
13533:∼
13432:×
13386:×
13362:⊗
13239:≅
13224:⊗
13095:⊗
13059:⊗
13014:
12999:
12984:⊗
12975:
12921:⊗
12916:∗
12897:∑
12893:↦
12880:⊗
12875:∗
12867:→
12825:" above:
12804:∗
12700:⊗
12647:⊗
12642:∗
12634:∈
12628:⊗
12576:≅
12570:⊗
12565:∗
12283:⟩
12271:⟨
12250:⊗
12204:→
12155:⟩
12141:∗
12127:⟨
12121:⟩
12100:⟨
12076:∗
12068:⊗
12047:transpose
12027:∗
12004:∈
11999:∗
11960:∗
11952:⊗
11946:−
11940:⊗
11916:⊗
11865:∈
11706:∗
11639:…
11597:∗
11584:⊗
11571:λ
11562:∑
11558:↦
11555:λ
11546:∗
11538:⊗
11532:→
11468:∗
11406:−
11395:−
11384:→
11272:↦
11266:⊗
11240:→
11235:∗
11227:⊗
11174:σ
11165:ν
11155:μ
11142:σ
11133:ν
11130:μ
11119:⊗
11091:γ
11081:β
11072:α
11059:γ
11050:β
11041:α
11029:⊗
10996:γ
10927:α
10922:β
10862:⋯
10794:⋯
10745:⋯
10714:⊗
10671:∈
10631:∈
10606:covariant
10525:∗
10469:⊗
10453:⊗
10418:⊗
10402:⊗
10252:→
10209:⊗
10130:∗
10097:⊗
10087:∗
10074:⊗
10066:⊗
10047:⏟
10041:∗
10033:⊗
10030:⋯
10027:⊗
10022:∗
10010:⊗
9999:⏟
9992:⊗
9989:⋯
9986:⊗
9853:
9847:×
9841:
9829:⊗
9823:
8683:⊗
8324:⊗
8298:⊗
8246:injective
8244:are both
8229:from the
8227:bifunctor
8197:⊗
8188:∘
8179:⊗
8161:⊗
8152:∘
8143:⊗
8128:⊗
8084:⊗
8060:⊗
8045:⊗
8010:⊗
8004:→
7998:⊗
7986:⊗
7955:→
7921:→
7886:⊗
7854:⊗
7830:⊗
7815:⊗
7780:⊗
7774:→
7768:⊗
7756:⊗
7718:→
7665:⊗
7657:→
7649:⊗
7600:⊗
7597:⋯
7594:⊗
7572:↦
7559:⊗
7556:⋯
7553:⊗
7496:⊗
7452:⊗
7426:⊗
7420:↦
7414:⊗
7383:⊗
7377:≠
7371:⊗
7308:⊗
7280:⊗
7248:⊗
7242:≅
7236:⊗
7147:⊗
7138:⊗
7110:⊗
7101:⊗
7063:⊗
7054:⊗
7048:≅
7042:⊗
7033:⊗
6941:⊗
6900:⊗
6887:dimension
6869:Dimension
6797:⊗
6788:↦
6754:→
6744:×
6708:×
6670:∈
6658:∈
6646:⊗
6635:
6594:⊆
6559:⊆
6510:↦
6468:×
6455:∈
6449:⊗
6412:∈
6375:∈
6295:↦
6220:↦
6114:×
6108:∈
6045:⊗
6021:⊗
5895:…
5872:…
5782:…
5740:…
5664:→
5649:×
5521:≤
5515:≤
5503:≤
5497:≤
5393:…
5327:…
5166:…
5091:then all
5062:…
4964:∑
4938:∈
4922:…
4886:∈
4870:…
4740:×
4590:→
4584:×
4481:∈
4453:∈
4424:⊗
4412:~
4364:⊗
4360:∘
4354:~
4317:→
4311:⊗
4299:~
4262:→
4256:×
4220:⊗
4192:×
4166:⊗
4160:↦
4138:⊗
4112:⊗
3965:⊗
3885:⊗
3847:∈
3819:∈
3765:∈
3686:−
3636:−
3582:−
3557:−
3486:−
3461:−
3297:→
3291:×
3279:functions
3260:∈
3232:∈
3163:×
3032:⊗
2995:⊗
2969:×
2943:⊗
2937:↦
2915:⊗
2754:⊗
2715:∈
2708:∑
2692:∈
2685:∑
2638:∈
2631:∑
2620:⊗
2584:∈
2577:∑
2556:⊗
2526:∈
2500:∈
2458:×
2397:×
2319:⊗
2280:∈
2273:∑
2257:∈
2250:∑
2220:⊗
2117:⊗
2065:∈
2053:∑
2037:∈
2025:∑
1997:⊗
1968:→
1962:×
1950:⊗
1917:×
1884:⊗
1849:×
1814:×
1808:∈
1700:∈
1693:∑
1677:∈
1670:∑
1525:∈
1518:∑
1475:∈
1468:∑
1342:→
1336:×
1302:×
1296:∈
1249:×
1214:×
1184:⊗
1091:⊗
1053:∈
1034:∈
1028:∣
1022:⊗
988:⊗
949:×
880:⊗
839:×
818:from the
816:functions
799:⊗
764:∈
729:∈
703:⊗
669:⊗
475:→
469:⊗
436:×
399:⊗
357:⊗
331:⊗
290:⊗
252:⊗
221:⊗
193:⊗
167:∈
155:∈
108:⊗
102:→
96:×
48:⊗
21050:Manifold
21035:Geodesic
20793:Notation
20555:(1999).
20510:(2002),
20488:(2003).
20466:(1974).
20437:Archived
20409:(1989).
20278:citation
20267:archived
20196:Archived
20040:See also
18595:algebras
17753:defines
17081:matrices
16957:, then:
16844:is some
16838:, where
16430: :
15961:cokernel
15820:-module
15814:of some
15112:for all
13850:′
13799:′
13760:′
13709:′
11012:. Then:
10604:are two
10465:′
10433:′
10286:′
10270:′
10237:′
10227:′
8285:matrices
8231:category
7403:The map
6034:so that
3186:are the
2144:′
2133:′
2104:′
2089:′
2061:′
2033:′
1785:for any
1008:The set
522:manifold
208:denoted
21147:Physics
20981:Related
20744:Physics
20662:Tensors
20557:Algebra
20534:1878556
20512:Algebra
20490:Algebra
20364:1468229
20231:Algebra
20046:Dyadics
20036:/APL).
20034:Fortran
20013:, then
19968:A ○.× B
19578:
19548:
19506:
19437:
19027:above.
19019:of the
18482:of the
18480:variety
18295:
18231:
17444:
17409:
17361:
17341:
17265:
17245:
17175:
17135:
17079:Square
17069:
17030:
16351:
16311:
16297:
16264:
16254:
16227:
16159:in the
16091:
16042:
15704:
15678:
15586:
15554:
15427:
15371:
15315:
15259:
15166:
15140:
14087:
14055:
14012:
13986:
13962:
13936:
13466:is the
13458:-module
13450:is the
13336:over a
13323:modules
12684:
12664:
12478:
12441:
12426:
12389:
12090:
12057:
12045:is the
11897:
11854:
11850:
11818:
11720:is its
11686:
11666:
11337:, with
11335:
11303:
11195:algebra
11010:
10983:
10941:
10909:
10698:
10660:
10590:tensors
10144:is the
9903:a type
9868:
9812:
8339:is the
7970:
7938:
7733:
7701:
7682:
7636:
7483:), let
7398:
7360:
7356:
7330:
7323:
7297:
7165:
7127:
7014:
6982:
6889:, then
6858:
6838:
6546:
6487:
6436:
6401:
6324:). Let
6322:
6284:
6129:
6085:
5990:
5955:
5440:
5420:
5240:
5220:
5145:if all
5140:
5120:
5041:if all
5035:
4955:
4764:
4719:
4545:
4519:
4512:Theorem
4496:
4470:
4332:
4285:
4277:
4239:
4235:
4209:
4127:
4101:
3980:
3954:
3862:
3836:
3780:
3728:
3391:be the
3275:
3249:
3135:over a
3131:be two
3108:over a
3093:
3073:
3047:is the
2904:
2877:
2480:
2440:
2412:
2386:
1871:
1831:
1593:'s and
1357:
1319:
1271:
1231:
1168:
1141:
1106:
1080:
1003:
977:
786:
753:
650:
623:
584:over a
580:be two
561:objects
236:
210:
63:of two
21075:Vector
21070:Spinor
21055:Matrix
20849:Tensor
20614:
20604:
20582:
20563:
20540:
20532:
20522:
20496:
20474:
20452:
20417:
20385:
20362:
20352:
20237:
20133:
20022:MATLAB
20015:a */ b
19984:a */ b
19974:). In
19604:-forms
19580:, the
19067:, the
19063:, the
19059:, the
18509:minors
17363:. The
17133:, say
16811:. The
16565:is an
15621:is a (
15489:is a (
15448:is a (
15436:is a (
15324:is a (
15212:is a (
14207:where
13110:where
11986:where
11688:, and
11618:where
11346:> 0
11211:(1, 1)
10905:(1, 1)
9936:tensor
9887:Tensor
8357:, and
7739:, the
6548:. If
6438:, let
4547:, and
4281:unique
4081:linear
3726:where
3137:field
1640:that:
555:; see
501:, the
307:tensor
149:
34:, the
20995:Basis
20680:Scope
20323:(PDF)
20299:arXiv
20270:(PDF)
20259:(PDF)
20082:Notes
18173:is a
17559:be a
17291:then
17267:. If
17243:over
17104:field
16947:is a
16439:with
16040:Here
15990:coker
15512:are (
14246:over
14242:is a
13677:is a
13452:free
11105:and:
10157:from
10117:Here
7441:from
5142:, and
4181:from
3221:with
3188:pairs
3180:basis
3178:as a
2416:basis
869:make
718:with
593:bases
586:field
542:up to
490:(see
374:bases
313:or a
305:is a
80:field
20612:OCLC
20602:ISBN
20580:ISBN
20561:ISBN
20520:ISBN
20494:ISBN
20472:ISBN
20450:ISBN
20415:ISBN
20383:ISBN
20350:ISBN
20284:link
20235:ISBN
20215:See
20156:and
20131:ISBN
20009:are
20005:and
19997:and
18678:and
18528:and
18327:rank
17796:and
16799:and
16523:are
16517:and
16499:Let
16225:(in
15552:and
15508:and
15138:and
14141:and
14121:Let
13343:ring
13330:and
13168:and
12542:Hom(
11455:and
10658:and
10614:and
10598:and
10354:and
9897:and
9800:rank
8369:and
8277:and
8240:and
7936:and
7198:and
6921:and
6879:and
6836:and
6632:span
6583:and
6399:and
6344:and
6259:and
5953:and
5565:and
5286:are
5266:and
5218:are
5118:are
5087:are
4901:and
4806:are
4786:and
4727:span
4657:and
4517:Let
4468:and
4095:and
4087:The
3834:and
3387:Let
3380:and
3247:and
3127:and
3123:Let
3110:ring
3071:and
2875:and
2515:and
1453:as:
1426:and
1379:and
1139:and
751:and
688:and
655:The
621:and
576:and
572:Let
536:The
382:and
319:span
275:and
72:and
23:and
20538:Zbl
19990:).
19986:or
19970:or
19964:○.×
19960:APL
19780:Sym
19590:of
19586:th
18229:in
17700:of
17486:is
17367:of
17319:of
17311:is
17287:is
17239:of
17129:of
17043:deg
16951:of
16192:to
15670:of
15500:If
15432:If
15320:If
15208:If
14020:of
13984:of
13197:all
13011:dim
12996:dim
12972:dim
12540:as
12534:to
12516:as
12510:of
12439:of
12433:ad(
12309:in
12049:of
11513:):
11505:is
10512:of
10496:If
10169:).
8268:).
8248:or
8236:If
7295:to
7218:is
7125:to
6873:If
6690:of
5817:to
5418:in
5352:in
5037:,
4207:to
3993:.)
3403:is
3395:of
3348:on
3142:.
3021:of
2984:to
2350:Hom
1506:and
1203:on
971:to
927:to
861:to
684:of
544:an
494:).
271:of
30:In
21293::
20610:.
20551:;
20536:,
20530:MR
20528:,
20514:,
20435:.
20431:.
20360:MR
20358:.
20340:.
20280:}}
20276:{{
20261:,
20190:.
20105:^
20090:^
19980:*/
19942:.
19795::=
19630::=
19613::
19606:.
19214::=
19131::=
18954:.
18601:.
18554:.
17071:.
16942:=
16828:/
16824:=
16485:.
16465:→
16434:→
16425:,
15963::
15568::=
15534::=
15397::=
15285::=
15168:.
14169::=
13579:.
13476:.
13374::=
13189:,
13183:,
13174:.
12686:,
12519:U*
12494:,
12480:.
12092:,
11899:,
11352::
11342:,
11201:.
10176::
9873:A
9870:.
9850:Tr
9838:Tr
9820:Tr
8352:,
8348:,
7325:.
7167:.
6968:.
6927:.
6860:.
6629::=
6131:.
6052::=
5966::=
5929::=
5596::=
4498:).
4053:).
4048:∘
4035:=
3982:.
3871::
3864:.
3782:,
3112:.
2021::=
1170:.
1005:.
652:.
422:.
238:.
20654:e
20647:t
20640:v
20626:.
20618:.
20588:.
20569:.
20502:.
20480:.
20458:.
20423:.
20391:.
20366:.
20325:.
20307:.
20301::
20286:)
20243:.
20219:.
20205:.
20158:y
20154:x
20139:.
20007:b
20003:a
19999:b
19995:a
19976:J
19924:)
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19909:v
19900:1
19897:+
19894:i
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19869:1
19866:+
19863:i
19859:v
19850:i
19846:v
19836:(
19831:/
19824:n
19814:V
19802:V
19792:V
19784:n
19757:.
19752:}
19745:2
19741:V
19734:)
19729:2
19725:v
19721:,
19716:1
19712:v
19708:(
19700:1
19696:v
19687:2
19683:v
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19651:{
19644:/
19639:V
19633:V
19627:V
19621:V
19601:n
19593:V
19583:n
19566:V
19561:n
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19540:(
19528:V
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19468:=
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19336:}
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19085:V
18928:.
18925:)
18920:m
18917:+
18914:k
18910:x
18906:,
18900:,
18895:1
18892:+
18889:k
18885:x
18881:(
18878:g
18875:)
18870:k
18866:x
18862:,
18856:,
18851:1
18847:x
18843:(
18840:f
18837:=
18834:)
18829:m
18826:+
18823:k
18819:x
18815:,
18809:,
18804:1
18800:x
18796:(
18793:)
18790:g
18784:f
18781:(
18761:K
18741:V
18721:)
18716:m
18712:x
18708:,
18702:,
18697:1
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18689:(
18686:g
18666:)
18661:k
18657:x
18653:,
18647:,
18642:1
18638:x
18634:(
18631:f
18496:2
18490:2
18466:1
18458:)
18452:)
18448:x
18444:(
18439:n
18424:)
18420:x
18416:(
18411:2
18401:)
18397:x
18393:(
18388:1
18374:n
18370:x
18357:2
18353:x
18345:1
18341:x
18334:(
18305:A
18282:)
18276:n
18272:x
18268:,
18262:,
18257:1
18253:x
18248:(
18244:=
18240:x
18217:1
18211:d
18189:i
18141:n
18119:n
18116:,
18110:,
18107:1
18104:=
18101:i
18085:d
18081:j
18076:x
18065:3
18061:j
18056:x
18048:2
18044:j
18039:x
18031:d
18027:j
18018:3
18014:j
18008:2
18004:j
18000:i
17996:a
17990:n
17985:1
17982:=
17977:d
17973:j
17959:n
17954:1
17951:=
17946:3
17942:j
17931:n
17926:1
17923:=
17918:2
17914:j
17905:=
17902:)
17897:n
17893:x
17889:,
17883:,
17878:1
17874:x
17870:(
17865:i
17838:1
17832:n
17827:P
17817:1
17811:n
17806:P
17782:n
17778:K
17769:n
17765:K
17739:d
17732:)
17726:n
17722:K
17718:(
17712:A
17688:K
17668:)
17661:d
17657:i
17648:2
17644:i
17638:1
17634:i
17629:a
17625:(
17605:n
17593:n
17587:n
17567:d
17547:)
17540:d
17536:i
17527:2
17523:i
17517:1
17513:i
17508:a
17504:(
17501:=
17498:A
17474:K
17454:A
17430:1
17424:n
17419:P
17395:n
17375:A
17327:A
17275:A
17253:K
17225:1
17219:n
17214:P
17204:1
17198:n
17193:P
17188::
17161:n
17157:K
17148:n
17144:K
17113:K
17090:A
17055:)
17052:f
17049:(
17039:A
17025:A
17010:)
17007:x
17004:(
17001:f
16997:/
16993:]
16990:x
16987:[
16984:A
16978:A
16973:R
16965:A
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15082:=
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14066:=
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13994:A
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13928:φ
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12268:=
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12124:=
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11925:=
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11875:d
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11801:V
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11147:=
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11113:(
11087:V
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11064:=
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10992:V
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10058:=
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9976:=
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9967:(
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9923:)
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9835:=
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9784:.
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8099:.
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8069:=
8066:)
8063:w
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8054:(
8051:)
8048:g
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8039:(
8013:Z
8007:V
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