4206:
information needed, given a particular subtree, to construct its parent tree. This information is a tuple that contains a binary indicator of whether the child is on the left or right, the value at the parent, and the sibling subtree. This type can be represented as 2×A×T, which looks very much like the derivative of the transformation that generated the tree type.
4873:
1651:), the Fréchet derivative corresponds to taking the derivative of each component separately. The resulting derivative can be mapped to a vector. This is useful, for example, if the vector-valued function is the position vector of a particle through time, then the derivative is the velocity vector of the particle through time.
5091:
Multiplicative calculus replaces addition with multiplication, and hence rather than dealing with the limit of a ratio of differences, it deals with the limit of an exponentiation of ratios. This allows the development of the geometric derivative and bigeometric derivative. Moreover, just like the
4201:
containing values of type A can be represented as the algebra generated by the transformation 1+A×T→T. The "1" represents the construction of an empty tree, and the second term represents the construction of a tree from a value and two subtrees. The "+" indicates that a tree can be constructed
4205:
The derivative of such a type is the type that describes the context of a particular substructure with respect to its next outer containing structure. Put another way, it is the type representing the "difference" between the two. In the tree example, the derivative is a type that describes the
4710:
4679:
3835:
1677:
satisfies a weaker form of the
Leibniz (product) rule. It specializes the Fréchet derivative to the objects of geometric algebra. Geometric calculus is a powerful formalism that has been shown to encompass the similar frameworks of differential forms and differential geometry.
4127:
2252:
2503:
2623:
In the real numbers one can iterate the differentiation process, that is, apply derivatives more than once, obtaining derivatives of second and higher order. Higher derivatives can also be defined for functions of several variables, studied in
3493:
4565:
3616:
1722:
are special cases of the exterior derivative. An intuitive interpretation of the gradient is that it points "up": in other words, it points in the direction of fastest increase of the function. It can be used to calculate
1310:
2648:. One of the subtle points is that the higher derivatives are not intrinsically defined, and depend on the choice of the coordinates in a complicated fashion (in particular, the Hessian matrix of a function is not a
1998:
4892:
are a set of differential operators that permit the construction of a differential calculus for complex functions that is entirely analogous to the ordinary differential calculus for functions of real variables.
2609:
4868:{\displaystyle \square ={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}.}
4425:
3579:
1434:
2080:
3191:
2143:
4439:
of zeroth, first and second order derivatives "all at once". This allows us to think of the set of solutions of this differential equation as a "generalized antiderivative" of its right hand side 4
136:
3906:
2374:
2342:
1928:
2138:
4302:
2971:
1488:
3325:
2999:
2932:
3377:
3372:
3077:
2824:
5418:
5336:
4511:
4161:
2279:
2542:
2722:
4346:
3043:
2366:
1186:
1081:
2615:, the dual space of test functions. Weak derivatives are particularly useful in the study of partial differential equations, and within parts of functional analysis.
1191:
1151:
5093:
3611:
1049:
2299:
2022:
1336:
1121:
1101:
4909:
defines the derivative with respect to a function of a functional on a space of functions. This is an extension of the directional derivative to an infinite
5065:
and others the derivative can be used to develop notions of smoothness, analycity, integration, Taylor series as well as a theory of differential equations.
1550:, specifying the rate of change of one range coordinate with respect to a change in a domain coordinate. Of course, the Jacobian matrix of the composition
4980:. Fréchet differentiability is a strictly stronger condition than Gateaux differentiability, even in finite dimensions. Between the two extremes is the
5329:
4177:
The notion of derivation applies to noncommutative as well as commutative rings, and even to non-associative algebraic structures, such as Lie algebras.
5216:, Garrett Sobczyk: Clifford Algebra to Geometric Calculus, a Unified Language for mathematics and Physics (Dordrecht/Boston:G.Reidel Publ.Co., 1984,
1769:(a degree -1 derivation on the exterior algebra defined by contraction with a vector field), the exterior derivative and the Lie derivative form a
5068:
It may be possible to combine two or more of the above different notions of extension or abstraction of the original derivative. For example, in
5057:(it is known that any local field of positive characteristic is isomorphic to a Laurent series field). Along with suitably defined analogs to the
4351:
3232:-analogues that were discovered in the 19th century, but remained relatively obscure for a big part of the 20th century, outside of the theory of
3502:
4674:{\displaystyle \Delta ={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}.}
1632:
takes into account changes due to time dependence and motion through space along a vector field, and is a special case of the total derivative.
5322:
1355:
1945:
3095:
1731:
functions or normal directions. Divergence gives a measure of how much "source" or "sink" near a point there is. It can be used to calculate
4229:
combines several derivatives, possibly of different orders, in one algebraic expression. This is especially useful in considering ordinary
3830:{\displaystyle \lim _{z\to x}{\frac {f(z)-f(x)}{z-x}}=\lim _{q\to 1}{\frac {f(qx)-f(x)}{qx-x}}=\lim _{q\to 1}{\frac {f(qx)-f(x)}{(q-1)x}}.}
232:
5754:
2551:
1750:
is the rate of change of a vector or tensor field along the flow of another vector field. On vector fields, it is an example of a
5643:
5498:
4999:, used for changing variables, to measures. It expresses one measure μ in terms of another measure ν (under certain conditions).
2247:{\textstyle D^{\alpha }\varphi :={\frac {\partial ^{|\alpha |}\varphi }{\partial x_{1}^{\alpha _{1}}\dotsm x_{n}^{\alpha _{n}}}}}
17:
2027:
5633:
5302:
4122:{\displaystyle \left(a_{d}x^{d}+a_{d-1}x^{d-1}+\cdots +a_{1}x+a_{0}\right)'=da_{d}x^{d-1}+(d-1)a_{d-1}x^{d-2}+\cdots +a_{1}.}
3262:
2498:{\displaystyle \int _{\mathbb {R} ^{n}}u\ D^{\alpha }\!\varphi \ dx=(-1)^{|\alpha |}\int _{\mathbb {R} ^{n}}v\ \varphi \ dx}
5390:
493:
473:
5423:
4688:
or wave operator is similar to the
Laplacian, but acts on functions of four variables. Its definition uses the indefinite
4450:
2304:
1890:
3329:
The q-derivative, the difference operator and the standard derivative can all be viewed as the same thing on different
969:
532:
55:
5221:
4951:. They can be used to define an analogue of exterior derivative from differential geometry that applies to arbitrary
3581:
The Hahn difference is not only a generalization of the q-derivative but also an extension of the forward difference.
2085:
488:
4197:
generated by a transformation that maps structures based on the type back into the type. For example, the type T of
2940:
478:
5035:
is an operation similar to usual differentiation but with the usual context of real or complex numbers changed to
5692:
5467:
4996:
4214:
3874:
1533:
809:
483:
463:
145:
5687:
5519:
3885:, but also turn up in many other areas, where they often agree with less algebraic definitions of derivatives.
2632:
with respect to different variables. For example, the second order partial derivatives of a scalar function of
2724:
are not commutative, the limit of the difference quotient yields two different derivatives: A left derivative
4516:
3866:
1876:
591:
538:
424:
5544:
4992:
4532:
4230:
2976:
2657:
1447:
250:
222:
5628:
2835:
333:
5536:
5147:
4885:, in much the same way that a normal derivative describes how a function is approximated by a linear map.
3336:
3048:
2730:
1931:
1315:
842:
455:
293:
265:
31:
5611:
5540:
3878:
2612:
1869:
1821:
1674:
1339:
713:
677:
459:
338:
227:
217:
2679:, there are various ways to define derivatives of fractional or negative orders, which are studied in
1824:
of a map between manifolds is the induced map between tangent spaces of those maps. It abstracts the
5682:
5595:
5524:
5238:
5162:
5111:
4244:
2511:
2257:
1942:. First define test functions, which are infinitely differentiable and compactly supported functions
318:
5616:
5503:
5349:
5040:
4977:
612:
177:
4881:
is a non-linear differential operator which describes how a complex function is approximated by a
2705:
5718:
4963:
4210:
4135:
1636:
1154:
926:
718:
607:
5621:
5450:
5368:
5194:
5180:
5153:
5141:
5092:
classical differential operator has a discrete analog, the difference operator, there are also
4914:
4307:
3584:
Also note that the q-derivative is nothing but a special case of the familiar derivative. Take
3004:
2625:
1865:
1814:
1802:
1724:
1494:
962:
891:
852:
736:
672:
596:
4940:
2351:
1159:
1054:
5510:
5440:
5363:
5345:
5117:
5105:
5081:
5044:
5010:
4906:
4889:
4882:
4878:
4226:
2696:
1837:
1782:
1629:
995:
991:
936:
602:
378:
323:
284:
190:
5428:
4523:
which assigns to each function its derivative is an example of a differential operator on a
3488:{\displaystyle {\frac {f(qx)-f(x)}{(q-1)x}}={\frac {f(x+\varepsilon )-f(x)}{\varepsilon }}.}
1130:
1019:
5638:
5552:
5493:
5385:
5267:
5185:
5168:
5085:
5058:
5025:
5018:
3889:
3870:
3256:
2001:
1939:
1841:
1801:
may be defined as a derivation at a point. This allows the abstraction of the notion of a
1759:
1728:
1659:
1529:
1521:
of a function. If such an operator exists, then it is unique, and can be represented by an
941:
921:
847:
516:
440:
414:
328:
3587:
8:
5674:
5664:
5547:
5462:
4952:
4936:
4902:
3330:
3252:
2680:
1857:
1703:
1699:
1622:
1028:
1023:
916:
886:
876:
763:
617:
419:
275:
158:
153:
5728:
5314:
5280:
5028:. There is no completely satisfactory analog of the first-order derivative or gradient.
5590:
5445:
5129:
5014:
4970:
4436:
4190:
3850:
2699:, derivatives can be defined in a similar way to real and complex functions. Since the
2629:
2284:
2007:
1736:
1715:
1691:
1670:
1547:
1321:
1106:
1086:
881:
784:
768:
708:
703:
698:
662:
543:
467:
373:
368:
172:
167:
2628:. In this case, instead of repeatedly applying the derivative, one repeatedly applies
5298:
5255:
5217:
4528:
4194:
4171:
3233:
2653:
2545:
955:
789:
567:
450:
403:
260:
255:
5733:
5580:
5570:
5472:
5433:
5247:
5135:
5069:
4981:
4948:
4944:
4685:
4543:
3225:
3221:
1853:
1770:
1766:
1687:
1655:
1648:
1614:
799:
693:
667:
528:
445:
409:
4962:, the usual definition of derivative is not quite strong enough, and one requires
5708:
5585:
5575:
5400:
5395:
5263:
5123:
4959:
4929:
4697:
4693:
4520:
4164:
1935:
1825:
1790:
1695:
1663:
1510:
1342:
931:
804:
758:
753:
640:
553:
498:
4241:) is a twice differentiable function of one variable, the differential equation
1305:{\displaystyle \lim _{\|h\|\to 0}{\frac {\|f(x+h)-f(x)-Ah\|_{W}}{\|h\|_{V}}}=0.}
5723:
5529:
5213:
4988:
4974:
4524:
4209:
This concept of a derivative of a type has practical applications, such as the
2937:
The existence of these limits are very restrictive conditions. For example, if
2684:
2652:). Nevertheless, higher derivatives have important applications to analysis of
2645:
1861:
1798:
1747:
1497:, in the context of differential equations defined by a vector valued function
814:
622:
394:
3845:
In algebra, generalizations of the derivative can be obtained by imposing the
1848:. This extends the directional derivative of scalar functions to sections of
5748:
5712:
5488:
5380:
5375:
5259:
5251:
5233:
4921:
4689:
3496:
3241:
3237:
3217:
2683:. The −1 order derivative corresponds to the integral, whence the term
1849:
1706:
and squares to zero. It is a grade 1 derivation on the exterior algebra. In
999:
794:
558:
313:
270:
4681:
Analogous operators can be defined for functions of any number of variables.
3873:
which satisfies the
Leibniz law (the product rule). Higher derivatives and
5659:
5514:
5174:
5077:
5048:
5036:
5003:
3846:
3089:
2665:
1786:
1610:
1606:
548:
298:
30:
This article is about the term as used in mathematics. For other uses, see
1338:, rather than at individual points, as not doing so tends to lead to many
5236:(1949). "Über Orthogonalpolynome, die q-Differenzengleichungen genügen".
4925:
4700:
4538:
Some of these operators are so important that they have their own names:
4198:
4186:
3854:
1844:
makes a choice for taking directional derivatives of vector fields along
1755:
1751:
983:
911:
2548:. This definition coincides with the classical derivative for functions
1993:{\displaystyle \varphi \in C_{c}^{\infty }\left(\mathbb {R} ^{n}\right)}
5410:
5062:
4555:
3893:
3877:
can also be defined. They are studied in a purely algebraic setting in
3220:. A large body of results from ordinary differential calculus, such as
2700:
1719:
1591:
1349:
987:
657:
581:
308:
303:
207:
5165: – Use of numerical analysis to estimate derivatives of functions
4515:
Combining derivatives of different variables results in a notion of a
4913:
vector space. An important case is the variational derivative in the
4910:
586:
576:
5562:
5080:. Thus one might want a derivative with some of the features of a
4559:
4444:
3882:
3082:
2661:
1805:
of a scalar function to general manifolds. For manifolds that are
1794:
1740:
1711:
1618:
1007:
652:
399:
356:
45:
5073:
5024:
Laplacians and differential equations using the
Laplacian can be
1517:
considered as a vector space over itself, and corresponds to the
1003:
5120: – Function defined on formal languages in computer science
2611:, and can be extended to a type of generalized functions called
3244:
have changed the situation dramatically, and the popularity of
2649:
2604:{\displaystyle u\in C^{|\alpha |}\left(\mathbb {R} ^{n}\right)}
1806:
1348:
The Fréchet derivative is quite similar to the formula for the
4429:
second order linear constant coefficient differential operator
2660:. For an advanced application of this analysis to topology of
1762:
of the manifold). It is a grade 0 derivation on the algebra.
994:
and admits many possible generalizations within the fields of
2973:
has left-derivatives at every point on an open connected set
1845:
4420:{\displaystyle L={\frac {d^{2}}{dx^{2}}}+2{\frac {d}{dx}}-3}
5419:
Differentiable vector–valued functions from
Euclidean space
3574:{\displaystyle {\frac {f(qx+\omega )-f(x)}{qx+\omega -x}}.}
1732:
1314:
Functions are defined as being differentiable in some open
4562:
of a scalar function of three variables, or explicitly as
2618:
5344:
5114: – Numerical calculations carrying along derivatives
1681:
1429:{\displaystyle \lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}=A,}
2075:{\displaystyle \alpha =(\alpha _{1},\dots ,\alpha _{n})}
1879:
extends the exterior derivative to vector valued forms.
5158:
Pages displaying short descriptions of redirect targets
3259:
is another discrete analog of the standard derivative.
3186:{\displaystyle D_{q}f(x)={\frac {f(qx)-f(x)}{(q-1)x}}.}
1860:, the existence of a metric chooses a unique preferred
1440:
to the left hand side. However, the Fréchet derivative
5283:, Robert S. Strichartz - Article in Notices of the AMS
2260:
2146:
2088:
1558:
is a product of corresponding
Jacobian matrices: J
5108: – Function defined on integers in number theory
4713:
4568:
4453:
4435:. The key idea here is that we consider a particular
4354:
4310:
4247:
4138:
3909:
3619:
3590:
3505:
3380:
3339:
3265:
3098:
3051:
3007:
2979:
2943:
2838:
2733:
2708:
2554:
2514:
2377:
2354:
2307:
2287:
2030:
2010:
1948:
1893:
1450:
1358:
1324:
1194:
1162:
1133:
1109:
1089:
1057:
1031:
58:
5190:
Pages displaying wikidata descriptions as a fallback
5094:
discrete analogs of these multiplicative derivatives
4535:
can be defined which allow for fractional calculus.
1934:, but not necessarily classically differentiable, a
1590:(ƒ). This is a higher-dimensional statement of the
5126: – Class of generalisations of the derivative
4867:
4673:
4505:
4419:
4340:
4296:
4155:
4121:
3829:
3605:
3573:
3487:
3366:
3319:
3185:
3071:
3037:
2993:
2965:
2926:
2818:
2716:
2603:
2536:
2497:
2360:
2337:{\displaystyle v:\mathbb {R} ^{n}\to \mathbb {R} }
2336:
2293:
2273:
2246:
2132:
2074:
2016:
1992:
1923:{\displaystyle u:\mathbb {R} ^{n}\to \mathbb {R} }
1922:
1482:
1428:
1330:
1304:
1180:
1145:
1115:
1095:
1075:
1043:
130:
2413:
27:Fundamental construction of differential calculus
5746:
5132: – Generalization of derivative to fractals
4550:is a second-order partial differential operator
3753:
3684:
3621:
3083:Difference operator, q-analogues and time scales
2840:
2735:
2133:{\textstyle |\alpha |:=\sum _{1}^{n}\alpha _{i}}
1360:
1196:
131:{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}
5171: – Type of derivative of a linear operator
5150: – Method of mathematical differentiation
2966:{\displaystyle f:\mathbb {H} \to \mathbb {H} }
1789:may be defined as a derivation on the ring of
5330:
2675: th derivatives for any natural number
963:
4233:with constant coefficients. For example, if
3212:we obtain the ordinary derivative, thus the
1662:, which are complex-valued functions on the
1284:
1277:
1266:
1220:
1206:
1200:
5177: – Q-analog of the ordinary derivative
2690:
1702:is the unique linear map which satisfies a
1609:), the Fréchet derivative corresponds to a
1352:found in elementary one-variable calculus,
5337:
5323:
5144: – Mathematical operation in calculus
5009:is a notion of derivative in the study of
3495:The q-derivative is a special case of the
1813:, this tangent vector will agree with the
970:
956:
4928:are generalizations of the derivative to
4443: − 1, by analogy with ordinary
4220:
3840:
3065:
2987:
2959:
2951:
2710:
2587:
2465:
2385:
2330:
2316:
1976:
1916:
1902:
1864:-free covariant derivative, known as the
1546:. Each entry of this matrix represents a
91:
5634:No infinite-dimensional Lebesgue measure
5297:. New York: Cambridge University Press.
5292:
5188: – generalization of the derivative
4896:
3092:of a function is defined by the formula
1776:
5644:Structure theorem for Gaussian measures
4180:
3320:{\displaystyle \Delta f(x)=f(x+1)-f(x)}
2619:Higher-order and fractional derivatives
1831:
494:Differentiating under the integral sign
14:
5747:
4170:. This definition can be extended to
1682:Exterior derivative and Lie derivative
5520:infinite-dimensional Gaussian measure
5318:
3849:in an algebraic structure, such as a
2994:{\displaystyle U\subset \mathbb {H} }
1872:for a treatment oriented to physics.
1666:where the Fréchet derivative exists.
1483:{\displaystyle t\mapsto f'(x)\cdot t}
1013:
5391:Infinite-dimensional vector function
5232:
4955:, instead of just smooth manifolds.
2927:{\displaystyle \lim _{h\to 0}\left.}
5295:Analysis in Positive Characteristic
3367:{\displaystyle \varepsilon =(q-1)x}
3072:{\displaystyle a,b\in \mathbb {H} }
2819:{\displaystyle \lim _{h\to 0}\left}
2636:variables can be organized into an
1882:
1743:" a vector field has near a point.
1658:, the central objects of study are
1022:defines the derivative for general
24:
4973:extends the Fréchet derivative to
4846:
4836:
4797:
4787:
4765:
4755:
4733:
4723:
4652:
4642:
4620:
4610:
4588:
4578:
4569:
4506:{\displaystyle f(x)=L^{-1}(4x-1).}
3266:
2191:
2167:
1965:
1621:but it is more natural to use the
40:Part of a series of articles about
25:
5766:
5755:Generalizations of the derivative
5458:Generalizations of the derivative
5424:Differentiation in Fréchet spaces
3216:-derivative may be viewed as its
1704:graded version of the Leibniz law
1617:. This can be interpreted as the
990:is a fundamental construction of
5072:, one studies spaces which look
4215:functional programming languages
3875:algebraic differential operators
3201:is a differentiable function of
2508:If such a function exists, then
2274:{\textstyle \alpha ^{\text{th}}}
5693:Holomorphic functional calculus
4943:are universal derivations of a
4297:{\displaystyle f''+2f'-3f=4x-1}
3847:Leibniz rule of differentiation
2537:{\displaystyle D^{\alpha }u:=v}
1597:For real valued functions from
5688:Continuous functional calculus
5286:
5274:
5226:
5207:
5017:. It is used in the study of
4497:
4482:
4463:
4457:
4320:
4314:
4304:may be rewritten in the form
4142:
4062:
4050:
3869:is a linear map on a ring or
3860:
3815:
3803:
3798:
3792:
3783:
3774:
3760:
3729:
3723:
3714:
3705:
3691:
3663:
3657:
3648:
3642:
3628:
3542:
3536:
3527:
3512:
3473:
3467:
3458:
3446:
3428:
3416:
3411:
3405:
3396:
3387:
3358:
3346:
3314:
3308:
3299:
3287:
3278:
3272:
3171:
3159:
3154:
3148:
3139:
3130:
3118:
3112:
3017:
3011:
2955:
2895:
2889:
2880:
2868:
2847:
2803:
2797:
2788:
2776:
2742:
2575:
2567:
2452:
2444:
2439:
2429:
2326:
2301:exists if there is a function
2180:
2172:
2098:
2090:
2069:
2037:
1912:
1542:(ƒ) of the mapping ƒ at point
1471:
1465:
1454:
1408:
1402:
1393:
1381:
1367:
1253:
1247:
1238:
1226:
1209:
1172:
1067:
125:
119:
110:
104:
88:
82:
13:
1:
5293:Kochubei, Anatoly N. (2009).
5156: – Branch of mathematics
4533:pseudo-differential operators
4517:partial differential operator
4231:linear differential equations
2140:. Applied to test functions,
1877:exterior covariant derivative
425:Integral of inverse functions
5138: – Mathematical concept
4163:is then a derivation on the
2717:{\displaystyle \mathbb {H} }
7:
5148:Logarithmic differentiation
5099:
4156:{\displaystyle f\mapsto f'}
1938:may be defined by means of
1822:differential or pushforward
843:Calculus on Euclidean space
266:Logarithmic differentiation
32:derivative (disambiguation)
10:
5771:
5047:with coefficients in some
3879:differential Galois theory
3248:-analogues is on the rise.
1870:gauge covariant derivative
1739:. Curl measures how much "
29:
5701:
5683:Borel functional calculus
5673:
5652:
5604:
5561:
5481:
5409:
5356:
5350:topological vector spaces
5239:Mathematische Nachrichten
5163:Numerical differentiation
5112:Automatic differentiation
4978:topological vector spaces
4932:used in convex analysis.
4341:{\displaystyle L(f)=4x-1}
3038:{\displaystyle f(q)=a+qb}
1519:best linear approximation
1505:, the Fréchet derivative
577:Summand limit (term test)
5617:Inverse function theorem
5504:Classical Wiener measure
5252:10.1002/mana.19490020103
5200:
4993:Radon–Nikodym derivative
4964:strict differentiability
4193:can be described as the
3896:over a commutative ring
2691:Quaternionic derivatives
2361:{\displaystyle \varphi }
1754:(vector fields form the
1181:{\displaystyle A:V\to W}
1076:{\displaystyle f:U\to W}
261:Implicit differentiation
251:Differentiation notation
178:Inverse function theorem
5719:Convenient vector space
4431:acting on functions of
2829:and a right derivative
1725:directional derivatives
1637:vector-valued functions
1155:bounded linear operator
719:Helmholtz decomposition
5612:Cameron–Martin theorem
5369:Classical Wiener space
5195:Topological derivative
5181:Semi-differentiability
5154:Non-classical analysis
5142:Logarithmic derivative
5011:abstract Wiener spaces
4915:calculus of variations
4869:
4675:
4507:
4447:, and formally write
4421:
4342:
4298:
4221:Differential operators
4157:
4123:
3841:Derivatives in algebra
3831:
3607:
3575:
3489:
3368:
3333:. For example, taking
3321:
3187:
3073:
3039:
2995:
2967:
2928:
2820:
2718:
2626:multivariable calculus
2605:
2538:
2499:
2362:
2338:
2295:
2275:
2248:
2134:
2119:
2076:
2018:
1994:
1924:
1866:Levi-Civita connection
1815:directional derivative
1803:directional derivative
1495:multivariable calculus
1484:
1430:
1332:
1306:
1182:
1147:
1146:{\displaystyle x\in U}
1125:Fréchet differentiable
1117:
1097:
1077:
1051:. Briefly, a function
1045:
853:Limit of distributions
673:Directional derivative
334:Faà di Bruno's formula
132:
18:Generalized derivative
5629:Feldman–Hájek theorem
5441:Functional derivative
5364:Abstract Wiener space
5118:Brzozowski derivative
5106:Arithmetic derivative
5082:functional derivative
5045:formal Laurent series
4907:functional derivative
4897:Other generalizations
4890:Wirtinger derivatives
4883:fractional-linear map
4879:Schwarzian derivative
4870:
4676:
4508:
4422:
4343:
4299:
4227:differential operator
4158:
4124:
3832:
3608:
3576:
3490:
3369:
3322:
3240:and the discovery of
3205:then in the limit as
3188:
3074:
3040:
2996:
2968:
2929:
2821:
2719:
2697:quaternionic analysis
2656:of a function at its
2606:
2539:
2500:
2363:
2339:
2296:
2276:
2249:
2135:
2105:
2077:
2019:
1995:
1925:
1838:differential geometry
1783:differential topology
1777:Differential topology
1660:holomorphic functions
1630:convective derivative
1485:
1444:denotes the function
1431:
1333:
1307:
1183:
1148:
1118:
1103:is an open subset of
1098:
1078:
1046:
996:mathematical analysis
992:differential calculus
937:Mathematical analysis
848:Generalized functions
533:arithmetico-geometric
379:Leibniz integral rule
133:
5553:Radonifying function
5494:Cylinder set measure
5386:Cylinder set measure
5281:Analysis on Fractals
5186:Symmetric derivative
5169:Pincherle derivative
5086:covariant derivative
5059:exponential function
5019:stochastic processes
4941:Kähler differentials
4711:
4566:
4451:
4352:
4308:
4245:
4181:Derivative of a type
4136:
3907:
3617:
3606:{\displaystyle z=qx}
3588:
3503:
3378:
3337:
3263:
3257:difference equations
3096:
3049:
3005:
2977:
2941:
2836:
2731:
2706:
2552:
2512:
2375:
2352:
2305:
2285:
2258:
2144:
2086:
2028:
2008:
1946:
1940:integration by parts
1891:
1842:covariant derivative
1832:Covariant derivative
1760:diffeomorphism group
1675:geometric derivative
1448:
1356:
1322:
1192:
1160:
1131:
1107:
1087:
1055:
1029:
1024:normed vector spaces
942:Nonstandard analysis
415:Lebesgue integration
285:Rules and identities
56:
5675:Functional calculus
5665:Covariance operator
5586:Gelfand–Pettis/Weak
5548:measurable function
5463:Hadamard derivative
5026:defined on fractals
4953:algebraic varieties
4937:commutative algebra
4903:functional analysis
4191:abstract data types
3253:difference operator
2681:fractional calculus
2630:partial derivatives
2281:weak derivative of
2240:
2215:
2004:, which are length
1969:
1858:Riemannian geometry
1700:exterior derivative
1623:exterior derivative
1044:{\displaystyle V,W}
613:Cauchy condensation
420:Contour integration
146:Fundamental theorem
73:
5622:Nash–Moser theorem
5499:Canonical Gaussian
5446:Gateaux derivative
5429:Fréchet derivative
5130:Fractal derivative
5033:Carlitz derivative
5015:Malliavin calculus
4971:Gateaux derivative
4865:
4671:
4527:. By means of the
4503:
4437:linear combination
4417:
4338:
4294:
4213:technique used in
4172:rational functions
4153:
4119:
3881:and the theory of
3827:
3767:
3698:
3635:
3603:
3571:
3485:
3364:
3317:
3236:. The progress of
3183:
3069:
3035:
2991:
2963:
2924:
2854:
2816:
2749:
2714:
2601:
2544:, which is unique
2534:
2495:
2358:
2334:
2291:
2271:
2244:
2219:
2194:
2130:
2072:
2024:lists of integers
2014:
1990:
1955:
1932:locally integrable
1920:
1765:Together with the
1737:divergence theorem
1692:differential forms
1671:geometric calculus
1548:partial derivative
1480:
1426:
1374:
1328:
1302:
1216:
1178:
1153:if there exists a
1143:
1113:
1093:
1073:
1041:
1020:Fréchet derivative
1014:Fréchet derivative
785:Partial derivative
714:generalized Stokes
608:Alternating series
489:Reduction formulae
464:tangent half-angle
451:Cylindrical shells
374:Integral transform
369:Lists of integrals
173:Mean value theorem
128:
59:
5742:
5741:
5639:Sazonov's theorem
5525:Projection-valued
5304:978-0-521-50977-0
4860:
4831:
4811:
4779:
4747:
4696:, instead of the
4666:
4634:
4602:
4529:Fourier transform
4409:
4388:
3890:formal derivative
3888:For example, the
3822:
3752:
3747:
3683:
3678:
3620:
3566:
3480:
3435:
3234:special functions
3178:
2839:
2734:
2546:almost everywhere
2488:
2482:
2419:
2402:
2294:{\displaystyle u}
2268:
2242:
2017:{\displaystyle n}
1887:Given a function
1854:principal bundles
1649:parametric curves
1436:and simply moves
1415:
1359:
1331:{\displaystyle x}
1294:
1195:
1116:{\displaystyle V}
1096:{\displaystyle U}
980:
979:
860:
859:
822:
821:
790:Multiple integral
726:
725:
630:
629:
597:Direct comparison
568:Convergence tests
506:
505:
479:Partial fractions
346:
345:
256:Second derivative
16:(Redirected from
5762:
5734:Hilbert manifold
5729:Fréchet manifold
5513: like
5473:Quasi-derivative
5339:
5332:
5325:
5316:
5315:
5309:
5308:
5290:
5284:
5278:
5272:
5271:
5230:
5224:
5211:
5191:
5159:
5136:Hasse derivative
5070:Finsler geometry
4995:generalizes the
4982:quasi-derivative
4945:commutative ring
4930:convex functions
4874:
4872:
4871:
4866:
4861:
4859:
4858:
4857:
4844:
4843:
4834:
4832:
4830:
4829:
4817:
4812:
4810:
4809:
4808:
4795:
4794:
4785:
4780:
4778:
4777:
4776:
4763:
4762:
4753:
4748:
4746:
4745:
4744:
4731:
4730:
4721:
4680:
4678:
4677:
4672:
4667:
4665:
4664:
4663:
4650:
4649:
4640:
4635:
4633:
4632:
4631:
4618:
4617:
4608:
4603:
4601:
4600:
4599:
4586:
4585:
4576:
4553:
4546:or Laplacian on
4544:Laplace operator
4512:
4510:
4509:
4504:
4481:
4480:
4426:
4424:
4423:
4418:
4410:
4408:
4397:
4389:
4387:
4386:
4385:
4372:
4371:
4362:
4347:
4345:
4344:
4339:
4303:
4301:
4300:
4295:
4269:
4255:
4162:
4160:
4159:
4154:
4152:
4128:
4126:
4125:
4120:
4115:
4114:
4096:
4095:
4080:
4079:
4046:
4045:
4030:
4029:
4014:
4010:
4006:
4005:
4004:
3989:
3988:
3970:
3969:
3954:
3953:
3935:
3934:
3925:
3924:
3836:
3834:
3833:
3828:
3823:
3821:
3801:
3769:
3766:
3748:
3746:
3732:
3700:
3697:
3679:
3677:
3666:
3637:
3634:
3613:. Then we have,
3612:
3610:
3609:
3604:
3580:
3578:
3577:
3572:
3567:
3565:
3545:
3507:
3494:
3492:
3491:
3486:
3481:
3476:
3441:
3436:
3434:
3414:
3382:
3373:
3371:
3370:
3365:
3326:
3324:
3323:
3318:
3226:Taylor expansion
3222:binomial formula
3211:
3192:
3190:
3189:
3184:
3179:
3177:
3157:
3125:
3108:
3107:
3078:
3076:
3075:
3070:
3068:
3044:
3042:
3041:
3036:
3000:
2998:
2997:
2992:
2990:
2972:
2970:
2969:
2964:
2962:
2954:
2933:
2931:
2930:
2925:
2920:
2916:
2915:
2914:
2902:
2898:
2853:
2825:
2823:
2822:
2817:
2815:
2811:
2810:
2806:
2767:
2766:
2748:
2723:
2721:
2720:
2715:
2713:
2610:
2608:
2607:
2602:
2600:
2596:
2595:
2590:
2580:
2579:
2578:
2570:
2543:
2541:
2540:
2535:
2524:
2523:
2504:
2502:
2501:
2496:
2486:
2480:
2476:
2475:
2474:
2473:
2468:
2457:
2456:
2455:
2447:
2417:
2412:
2411:
2400:
2396:
2395:
2394:
2393:
2388:
2367:
2365:
2364:
2359:
2343:
2341:
2340:
2335:
2333:
2325:
2324:
2319:
2300:
2298:
2297:
2292:
2280:
2278:
2277:
2272:
2270:
2269:
2266:
2253:
2251:
2250:
2245:
2243:
2241:
2239:
2238:
2237:
2227:
2214:
2213:
2212:
2202:
2189:
2185:
2184:
2183:
2175:
2164:
2156:
2155:
2139:
2137:
2136:
2131:
2129:
2128:
2118:
2113:
2101:
2093:
2081:
2079:
2078:
2073:
2068:
2067:
2049:
2048:
2023:
2021:
2020:
2015:
1999:
1997:
1996:
1991:
1989:
1985:
1984:
1979:
1968:
1963:
1929:
1927:
1926:
1921:
1919:
1911:
1910:
1905:
1883:Weak derivatives
1791:smooth functions
1771:Lie superalgebra
1767:interior product
1688:exterior algebra
1656:complex analysis
1615:total derivative
1489:
1487:
1486:
1481:
1464:
1435:
1433:
1432:
1427:
1416:
1411:
1376:
1373:
1337:
1335:
1334:
1329:
1311:
1309:
1308:
1303:
1295:
1293:
1292:
1291:
1275:
1274:
1273:
1218:
1215:
1187:
1185:
1184:
1179:
1152:
1150:
1149:
1144:
1122:
1120:
1119:
1114:
1102:
1100:
1099:
1094:
1082:
1080:
1079:
1074:
1050:
1048:
1047:
1042:
972:
965:
958:
906:
871:
837:
836:
833:
800:Surface integral
743:
742:
739:
647:
646:
643:
603:Limit comparison
523:
522:
519:
410:Riemann integral
363:
362:
359:
319:L'Hôpital's rule
276:Taylor's theorem
197:
196:
193:
137:
135:
134:
129:
81:
72:
67:
37:
36:
21:
5770:
5769:
5765:
5764:
5763:
5761:
5760:
5759:
5745:
5744:
5743:
5738:
5709:Banach manifold
5697:
5669:
5648:
5600:
5576:Direct integral
5557:
5477:
5405:
5401:Vector calculus
5396:Matrix calculus
5352:
5343:
5313:
5312:
5305:
5291:
5287:
5279:
5275:
5231:
5227:
5212:
5208:
5203:
5189:
5157:
5124:Dini derivative
5102:
5056:
5043:in the form of
4960:p-adic analysis
4899:
4853:
4849:
4845:
4839:
4835:
4833:
4825:
4821:
4816:
4804:
4800:
4796:
4790:
4786:
4784:
4772:
4768:
4764:
4758:
4754:
4752:
4740:
4736:
4732:
4726:
4722:
4720:
4712:
4709:
4708:
4694:Minkowski space
4659:
4655:
4651:
4645:
4641:
4639:
4627:
4623:
4619:
4613:
4609:
4607:
4595:
4591:
4587:
4581:
4577:
4575:
4567:
4564:
4563:
4551:
4521:linear operator
4473:
4469:
4452:
4449:
4448:
4401:
4396:
4381:
4377:
4373:
4367:
4363:
4361:
4353:
4350:
4349:
4309:
4306:
4305:
4262:
4248:
4246:
4243:
4242:
4223:
4183:
4165:polynomial ring
4145:
4137:
4134:
4133:
4110:
4106:
4085:
4081:
4069:
4065:
4035:
4031:
4025:
4021:
4000:
3996:
3984:
3980:
3959:
3955:
3943:
3939:
3930:
3926:
3920:
3916:
3915:
3911:
3910:
3908:
3905:
3904:
3863:
3843:
3802:
3770:
3768:
3756:
3733:
3701:
3699:
3687:
3667:
3638:
3636:
3624:
3618:
3615:
3614:
3589:
3586:
3585:
3546:
3508:
3506:
3504:
3501:
3500:
3442:
3440:
3415:
3383:
3381:
3379:
3376:
3375:
3338:
3335:
3334:
3264:
3261:
3260:
3228:, have natural
3206:
3158:
3126:
3124:
3103:
3099:
3097:
3094:
3093:
3085:
3064:
3050:
3047:
3046:
3006:
3003:
3002:
2986:
2978:
2975:
2974:
2958:
2950:
2942:
2939:
2938:
2907:
2903:
2864:
2860:
2859:
2855:
2843:
2837:
2834:
2833:
2772:
2768:
2759:
2755:
2754:
2750:
2738:
2732:
2729:
2728:
2709:
2707:
2704:
2703:
2693:
2671:In addition to
2658:critical points
2621:
2591:
2586:
2585:
2581:
2574:
2566:
2565:
2561:
2553:
2550:
2549:
2519:
2515:
2513:
2510:
2509:
2469:
2464:
2463:
2462:
2458:
2451:
2443:
2442:
2438:
2407:
2403:
2389:
2384:
2383:
2382:
2378:
2376:
2373:
2372:
2353:
2350:
2349:
2348:test functions
2329:
2320:
2315:
2314:
2306:
2303:
2302:
2286:
2283:
2282:
2265:
2261:
2259:
2256:
2255:
2233:
2229:
2228:
2223:
2208:
2204:
2203:
2198:
2190:
2179:
2171:
2170:
2166:
2165:
2163:
2151:
2147:
2145:
2142:
2141:
2124:
2120:
2114:
2109:
2097:
2089:
2087:
2084:
2083:
2063:
2059:
2044:
2040:
2029:
2026:
2025:
2009:
2006:
2005:
1980:
1975:
1974:
1970:
1964:
1959:
1947:
1944:
1943:
1936:weak derivative
1915:
1906:
1901:
1900:
1892:
1889:
1888:
1885:
1834:
1826:Jacobian matrix
1779:
1696:smooth manifold
1684:
1664:complex numbers
1589:
1579:
1569:
1563:
1555:
1541:
1534:Jacobian matrix
1511:linear operator
1457:
1449:
1446:
1445:
1377:
1375:
1363:
1357:
1354:
1353:
1343:counterexamples
1323:
1320:
1319:
1287:
1283:
1276:
1269:
1265:
1219:
1217:
1199:
1193:
1190:
1189:
1161:
1158:
1157:
1132:
1129:
1128:
1108:
1105:
1104:
1088:
1085:
1084:
1056:
1053:
1052:
1030:
1027:
1026:
1016:
976:
947:
946:
932:Integration Bee
907:
904:
897:
896:
872:
869:
862:
861:
834:
831:
824:
823:
805:Volume integral
740:
735:
728:
727:
644:
639:
632:
631:
601:
520:
515:
508:
507:
499:Risch algorithm
474:Euler's formula
360:
355:
348:
347:
329:General Leibniz
212:generalizations
194:
189:
182:
168:Rolle's theorem
163:
138:
74:
68:
63:
57:
54:
53:
35:
28:
23:
22:
15:
12:
11:
5:
5768:
5758:
5757:
5740:
5739:
5737:
5736:
5731:
5726:
5724:Choquet theory
5721:
5716:
5705:
5703:
5699:
5698:
5696:
5695:
5690:
5685:
5679:
5677:
5671:
5670:
5668:
5667:
5662:
5656:
5654:
5650:
5649:
5647:
5646:
5641:
5636:
5631:
5626:
5625:
5624:
5614:
5608:
5606:
5602:
5601:
5599:
5598:
5593:
5588:
5583:
5578:
5573:
5567:
5565:
5559:
5558:
5556:
5555:
5550:
5534:
5533:
5532:
5527:
5522:
5508:
5507:
5506:
5501:
5491:
5485:
5483:
5479:
5478:
5476:
5475:
5470:
5465:
5460:
5455:
5454:
5453:
5443:
5438:
5437:
5436:
5426:
5421:
5415:
5413:
5407:
5406:
5404:
5403:
5398:
5393:
5388:
5383:
5378:
5373:
5372:
5371:
5360:
5358:
5357:Basic concepts
5354:
5353:
5342:
5341:
5334:
5327:
5319:
5311:
5310:
5303:
5285:
5273:
5234:Hahn, Wolfgang
5225:
5214:David Hestenes
5205:
5204:
5202:
5199:
5198:
5197:
5192:
5183:
5178:
5172:
5166:
5160:
5151:
5145:
5139:
5133:
5127:
5121:
5115:
5109:
5101:
5098:
5052:
5041:characteristic
4989:measure theory
4975:locally convex
4898:
4895:
4894:
4893:
4886:
4875:
4864:
4856:
4852:
4848:
4842:
4838:
4828:
4824:
4820:
4815:
4807:
4803:
4799:
4793:
4789:
4783:
4775:
4771:
4767:
4761:
4757:
4751:
4743:
4739:
4735:
4729:
4725:
4719:
4716:
4682:
4670:
4662:
4658:
4654:
4648:
4644:
4638:
4630:
4626:
4622:
4616:
4612:
4606:
4598:
4594:
4590:
4584:
4580:
4574:
4571:
4554:given by the
4525:function space
4502:
4499:
4496:
4493:
4490:
4487:
4484:
4479:
4476:
4472:
4468:
4465:
4462:
4459:
4456:
4416:
4413:
4407:
4404:
4400:
4395:
4392:
4384:
4380:
4376:
4370:
4366:
4360:
4357:
4337:
4334:
4331:
4328:
4325:
4322:
4319:
4316:
4313:
4293:
4290:
4287:
4284:
4281:
4278:
4275:
4272:
4268:
4265:
4261:
4258:
4254:
4251:
4222:
4219:
4182:
4179:
4151:
4148:
4144:
4141:
4130:
4129:
4118:
4113:
4109:
4105:
4102:
4099:
4094:
4091:
4088:
4084:
4078:
4075:
4072:
4068:
4064:
4061:
4058:
4055:
4052:
4049:
4044:
4041:
4038:
4034:
4028:
4024:
4020:
4017:
4013:
4009:
4003:
3999:
3995:
3992:
3987:
3983:
3979:
3976:
3973:
3968:
3965:
3962:
3958:
3952:
3949:
3946:
3942:
3938:
3933:
3929:
3923:
3919:
3914:
3900:is defined by
3862:
3859:
3842:
3839:
3838:
3837:
3826:
3820:
3817:
3814:
3811:
3808:
3805:
3800:
3797:
3794:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3765:
3762:
3759:
3755:
3751:
3745:
3742:
3739:
3736:
3731:
3728:
3725:
3722:
3719:
3716:
3713:
3710:
3707:
3704:
3696:
3693:
3690:
3686:
3682:
3676:
3673:
3670:
3665:
3662:
3659:
3656:
3653:
3650:
3647:
3644:
3641:
3633:
3630:
3627:
3623:
3602:
3599:
3596:
3593:
3582:
3570:
3564:
3561:
3558:
3555:
3552:
3549:
3544:
3541:
3538:
3535:
3532:
3529:
3526:
3523:
3520:
3517:
3514:
3511:
3484:
3479:
3475:
3472:
3469:
3466:
3463:
3460:
3457:
3454:
3451:
3448:
3445:
3439:
3433:
3430:
3427:
3424:
3421:
3418:
3413:
3410:
3407:
3404:
3401:
3398:
3395:
3392:
3389:
3386:
3374:, we may have
3363:
3360:
3357:
3354:
3351:
3348:
3345:
3342:
3327:
3316:
3313:
3310:
3307:
3304:
3301:
3298:
3295:
3292:
3289:
3286:
3283:
3280:
3277:
3274:
3271:
3268:
3249:
3242:quantum groups
3182:
3176:
3173:
3170:
3167:
3164:
3161:
3156:
3153:
3150:
3147:
3144:
3141:
3138:
3135:
3132:
3129:
3123:
3120:
3117:
3114:
3111:
3106:
3102:
3084:
3081:
3067:
3063:
3060:
3057:
3054:
3034:
3031:
3028:
3025:
3022:
3019:
3016:
3013:
3010:
2989:
2985:
2982:
2961:
2957:
2953:
2949:
2946:
2935:
2934:
2923:
2919:
2913:
2910:
2906:
2901:
2897:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2867:
2863:
2858:
2852:
2849:
2846:
2842:
2827:
2826:
2814:
2809:
2805:
2802:
2799:
2796:
2793:
2790:
2787:
2784:
2781:
2778:
2775:
2771:
2765:
2762:
2758:
2753:
2747:
2744:
2741:
2737:
2712:
2692:
2689:
2685:differintegral
2646:Hessian matrix
2620:
2617:
2599:
2594:
2589:
2584:
2577:
2573:
2569:
2564:
2560:
2557:
2533:
2530:
2527:
2522:
2518:
2506:
2505:
2494:
2491:
2485:
2479:
2472:
2467:
2461:
2454:
2450:
2446:
2441:
2437:
2434:
2431:
2428:
2425:
2422:
2416:
2410:
2406:
2399:
2392:
2387:
2381:
2357:
2344:such that for
2332:
2328:
2323:
2318:
2313:
2310:
2290:
2264:
2236:
2232:
2226:
2222:
2218:
2211:
2207:
2201:
2197:
2193:
2188:
2182:
2178:
2174:
2169:
2162:
2159:
2154:
2150:
2127:
2123:
2117:
2112:
2108:
2104:
2100:
2096:
2092:
2071:
2066:
2062:
2058:
2055:
2052:
2047:
2043:
2039:
2036:
2033:
2013:
1988:
1983:
1978:
1973:
1967:
1962:
1958:
1954:
1951:
1918:
1914:
1909:
1904:
1899:
1896:
1884:
1881:
1850:vector bundles
1833:
1830:
1799:tangent vector
1778:
1775:
1748:Lie derivative
1683:
1680:
1585:
1573:
1567:
1559:
1553:
1537:
1479:
1476:
1473:
1470:
1467:
1463:
1460:
1456:
1453:
1425:
1422:
1419:
1414:
1410:
1407:
1404:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1372:
1369:
1366:
1362:
1327:
1301:
1298:
1290:
1286:
1282:
1279:
1272:
1268:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1214:
1211:
1208:
1205:
1202:
1198:
1177:
1174:
1171:
1168:
1165:
1142:
1139:
1136:
1112:
1092:
1072:
1069:
1066:
1063:
1060:
1040:
1037:
1034:
1015:
1012:
978:
977:
975:
974:
967:
960:
952:
949:
948:
945:
944:
939:
934:
929:
927:List of topics
924:
919:
914:
908:
903:
902:
899:
898:
895:
894:
889:
884:
879:
873:
868:
867:
864:
863:
858:
857:
856:
855:
850:
845:
835:
830:
829:
826:
825:
820:
819:
818:
817:
812:
807:
802:
797:
792:
787:
779:
778:
774:
773:
772:
771:
766:
761:
756:
748:
747:
741:
734:
733:
730:
729:
724:
723:
722:
721:
716:
711:
706:
701:
696:
688:
687:
683:
682:
681:
680:
675:
670:
665:
660:
655:
645:
638:
637:
634:
633:
628:
627:
626:
625:
620:
615:
610:
605:
599:
594:
589:
584:
579:
571:
570:
564:
563:
562:
561:
556:
551:
546:
541:
536:
521:
514:
513:
510:
509:
504:
503:
502:
501:
496:
491:
486:
484:Changing order
481:
476:
471:
453:
448:
443:
435:
434:
433:Integration by
430:
429:
428:
427:
422:
417:
412:
407:
397:
395:Antiderivative
389:
388:
384:
383:
382:
381:
376:
371:
361:
354:
353:
350:
349:
344:
343:
342:
341:
336:
331:
326:
321:
316:
311:
306:
301:
296:
288:
287:
281:
280:
279:
278:
273:
268:
263:
258:
253:
245:
244:
240:
239:
238:
237:
236:
235:
230:
225:
215:
202:
201:
195:
188:
187:
184:
183:
181:
180:
175:
170:
164:
162:
161:
156:
150:
149:
148:
140:
139:
127:
124:
121:
118:
115:
112:
109:
106:
103:
100:
97:
94:
90:
87:
84:
80:
77:
71:
66:
62:
52:
49:
48:
42:
41:
26:
9:
6:
4:
3:
2:
5767:
5756:
5753:
5752:
5750:
5735:
5732:
5730:
5727:
5725:
5722:
5720:
5717:
5714:
5710:
5707:
5706:
5704:
5700:
5694:
5691:
5689:
5686:
5684:
5681:
5680:
5678:
5676:
5672:
5666:
5663:
5661:
5658:
5657:
5655:
5651:
5645:
5642:
5640:
5637:
5635:
5632:
5630:
5627:
5623:
5620:
5619:
5618:
5615:
5613:
5610:
5609:
5607:
5603:
5597:
5594:
5592:
5589:
5587:
5584:
5582:
5579:
5577:
5574:
5572:
5569:
5568:
5566:
5564:
5560:
5554:
5551:
5549:
5546:
5542:
5538:
5535:
5531:
5528:
5526:
5523:
5521:
5518:
5517:
5516:
5515:set functions
5512:
5509:
5505:
5502:
5500:
5497:
5496:
5495:
5492:
5490:
5489:Besov measure
5487:
5486:
5484:
5482:Measurability
5480:
5474:
5471:
5469:
5466:
5464:
5461:
5459:
5456:
5452:
5449:
5448:
5447:
5444:
5442:
5439:
5435:
5432:
5431:
5430:
5427:
5425:
5422:
5420:
5417:
5416:
5414:
5412:
5408:
5402:
5399:
5397:
5394:
5392:
5389:
5387:
5384:
5382:
5381:Convex series
5379:
5377:
5376:Bochner space
5374:
5370:
5367:
5366:
5365:
5362:
5361:
5359:
5355:
5351:
5347:
5340:
5335:
5333:
5328:
5326:
5321:
5320:
5317:
5306:
5300:
5296:
5289:
5282:
5277:
5269:
5265:
5261:
5257:
5253:
5249:
5246:(1–2): 4–34.
5245:
5241:
5240:
5235:
5229:
5223:
5222:90-277-2561-6
5219:
5215:
5210:
5206:
5196:
5193:
5187:
5184:
5182:
5179:
5176:
5173:
5170:
5167:
5164:
5161:
5155:
5152:
5149:
5146:
5143:
5140:
5137:
5134:
5131:
5128:
5125:
5122:
5119:
5116:
5113:
5110:
5107:
5104:
5103:
5097:
5095:
5089:
5087:
5083:
5079:
5078:Banach spaces
5075:
5071:
5066:
5064:
5060:
5055:
5050:
5046:
5042:
5038:
5034:
5029:
5027:
5022:
5020:
5016:
5012:
5008:
5006:
5000:
4998:
4994:
4990:
4985:
4983:
4979:
4976:
4972:
4967:
4965:
4961:
4956:
4954:
4950:
4946:
4942:
4938:
4933:
4931:
4927:
4923:
4922:subderivative
4918:
4916:
4912:
4908:
4904:
4891:
4887:
4884:
4880:
4876:
4862:
4854:
4850:
4840:
4826:
4822:
4818:
4813:
4805:
4801:
4791:
4781:
4773:
4769:
4759:
4749:
4741:
4737:
4727:
4717:
4714:
4706:
4702:
4699:
4695:
4691:
4690:metric tensor
4687:
4686:d'Alembertian
4683:
4668:
4660:
4656:
4646:
4636:
4628:
4624:
4614:
4604:
4596:
4592:
4582:
4572:
4561:
4557:
4549:
4545:
4541:
4540:
4539:
4536:
4534:
4530:
4526:
4522:
4518:
4513:
4500:
4494:
4491:
4488:
4485:
4477:
4474:
4470:
4466:
4460:
4454:
4446:
4442:
4438:
4434:
4430:
4414:
4411:
4405:
4402:
4398:
4393:
4390:
4382:
4378:
4374:
4368:
4364:
4358:
4355:
4335:
4332:
4329:
4326:
4323:
4317:
4311:
4291:
4288:
4285:
4282:
4279:
4276:
4273:
4270:
4266:
4263:
4259:
4256:
4252:
4249:
4240:
4236:
4232:
4228:
4218:
4216:
4212:
4207:
4203:
4200:
4196:
4192:
4188:
4178:
4175:
4173:
4169:
4166:
4149:
4146:
4139:
4116:
4111:
4107:
4103:
4100:
4097:
4092:
4089:
4086:
4082:
4076:
4073:
4070:
4066:
4059:
4056:
4053:
4047:
4042:
4039:
4036:
4032:
4026:
4022:
4018:
4015:
4011:
4007:
4001:
3997:
3993:
3990:
3985:
3981:
3977:
3974:
3971:
3966:
3963:
3960:
3956:
3950:
3947:
3944:
3940:
3936:
3931:
3927:
3921:
3917:
3912:
3903:
3902:
3901:
3899:
3895:
3891:
3886:
3884:
3880:
3876:
3872:
3868:
3858:
3856:
3852:
3848:
3824:
3818:
3812:
3809:
3806:
3795:
3789:
3786:
3780:
3777:
3771:
3763:
3757:
3749:
3743:
3740:
3737:
3734:
3726:
3720:
3717:
3711:
3708:
3702:
3694:
3688:
3680:
3674:
3671:
3668:
3660:
3654:
3651:
3645:
3639:
3631:
3625:
3600:
3597:
3594:
3591:
3583:
3568:
3562:
3559:
3556:
3553:
3550:
3547:
3539:
3533:
3530:
3524:
3521:
3518:
3515:
3509:
3498:
3482:
3477:
3470:
3464:
3461:
3455:
3452:
3449:
3443:
3437:
3431:
3425:
3422:
3419:
3408:
3402:
3399:
3393:
3390:
3384:
3361:
3355:
3352:
3349:
3343:
3340:
3332:
3328:
3311:
3305:
3302:
3296:
3293:
3290:
3284:
3281:
3275:
3269:
3258:
3254:
3250:
3247:
3243:
3239:
3238:combinatorics
3235:
3231:
3227:
3223:
3219:
3218:q-deformation
3215:
3209:
3204:
3200:
3196:
3180:
3174:
3168:
3165:
3162:
3151:
3145:
3142:
3136:
3133:
3127:
3121:
3115:
3109:
3104:
3100:
3091:
3087:
3086:
3080:
3061:
3058:
3055:
3052:
3032:
3029:
3026:
3023:
3020:
3014:
3008:
2983:
2980:
2947:
2944:
2921:
2917:
2911:
2908:
2904:
2899:
2892:
2886:
2883:
2877:
2874:
2871:
2865:
2861:
2856:
2850:
2844:
2832:
2831:
2830:
2812:
2807:
2800:
2794:
2791:
2785:
2782:
2779:
2773:
2769:
2763:
2760:
2756:
2751:
2745:
2739:
2727:
2726:
2725:
2702:
2698:
2688:
2686:
2682:
2678:
2674:
2669:
2667:
2663:
2659:
2655:
2654:local extrema
2651:
2647:
2643:
2639:
2635:
2631:
2627:
2616:
2614:
2613:distributions
2597:
2592:
2582:
2571:
2562:
2558:
2555:
2547:
2531:
2528:
2525:
2520:
2516:
2492:
2489:
2483:
2477:
2470:
2459:
2448:
2435:
2432:
2426:
2423:
2420:
2414:
2408:
2404:
2397:
2390:
2379:
2371:
2370:
2369:
2355:
2347:
2321:
2311:
2308:
2288:
2262:
2234:
2230:
2224:
2220:
2216:
2209:
2205:
2199:
2195:
2186:
2176:
2160:
2157:
2152:
2148:
2125:
2121:
2115:
2110:
2106:
2102:
2094:
2064:
2060:
2056:
2053:
2050:
2045:
2041:
2034:
2031:
2011:
2003:
2002:multi-indices
1986:
1981:
1971:
1960:
1956:
1952:
1949:
1941:
1937:
1933:
1907:
1897:
1894:
1880:
1878:
1873:
1871:
1867:
1863:
1859:
1855:
1851:
1847:
1843:
1839:
1829:
1827:
1823:
1818:
1816:
1812:
1808:
1804:
1800:
1796:
1792:
1788:
1784:
1774:
1772:
1768:
1763:
1761:
1757:
1753:
1749:
1744:
1742:
1738:
1734:
1730:
1726:
1721:
1717:
1713:
1709:
1705:
1701:
1697:
1693:
1689:
1679:
1676:
1672:
1667:
1665:
1661:
1657:
1652:
1650:
1646:
1642:
1638:
1633:
1631:
1626:
1624:
1620:
1616:
1612:
1608:
1607:scalar fields
1604:
1600:
1595:
1593:
1588:
1583:
1577:
1571:
1562:
1557:
1549:
1545:
1540:
1535:
1532:known as the
1531:
1528:
1524:
1520:
1516:
1512:
1508:
1504:
1500:
1496:
1491:
1477:
1474:
1468:
1461:
1458:
1451:
1443:
1439:
1423:
1420:
1417:
1412:
1405:
1399:
1396:
1390:
1387:
1384:
1378:
1370:
1364:
1351:
1346:
1344:
1341:
1325:
1317:
1316:neighbourhood
1312:
1299:
1296:
1288:
1280:
1270:
1262:
1259:
1256:
1250:
1244:
1241:
1235:
1232:
1229:
1223:
1212:
1203:
1175:
1169:
1166:
1163:
1156:
1140:
1137:
1134:
1126:
1110:
1090:
1070:
1064:
1061:
1058:
1038:
1035:
1032:
1025:
1021:
1011:
1009:
1005:
1001:
1000:combinatorics
997:
993:
989:
985:
973:
968:
966:
961:
959:
954:
953:
951:
950:
943:
940:
938:
935:
933:
930:
928:
925:
923:
920:
918:
915:
913:
910:
909:
901:
900:
893:
890:
888:
885:
883:
880:
878:
875:
874:
866:
865:
854:
851:
849:
846:
844:
841:
840:
839:
838:
828:
827:
816:
813:
811:
808:
806:
803:
801:
798:
796:
795:Line integral
793:
791:
788:
786:
783:
782:
781:
780:
776:
775:
770:
767:
765:
762:
760:
757:
755:
752:
751:
750:
749:
745:
744:
738:
737:Multivariable
732:
731:
720:
717:
715:
712:
710:
707:
705:
702:
700:
697:
695:
692:
691:
690:
689:
685:
684:
679:
676:
674:
671:
669:
666:
664:
661:
659:
656:
654:
651:
650:
649:
648:
642:
636:
635:
624:
621:
619:
616:
614:
611:
609:
606:
604:
600:
598:
595:
593:
590:
588:
585:
583:
580:
578:
575:
574:
573:
572:
569:
566:
565:
560:
557:
555:
552:
550:
547:
545:
542:
540:
537:
534:
530:
527:
526:
525:
524:
518:
512:
511:
500:
497:
495:
492:
490:
487:
485:
482:
480:
477:
475:
472:
469:
465:
461:
460:trigonometric
457:
454:
452:
449:
447:
444:
442:
439:
438:
437:
436:
432:
431:
426:
423:
421:
418:
416:
413:
411:
408:
405:
401:
398:
396:
393:
392:
391:
390:
386:
385:
380:
377:
375:
372:
370:
367:
366:
365:
364:
358:
352:
351:
340:
337:
335:
332:
330:
327:
325:
322:
320:
317:
315:
312:
310:
307:
305:
302:
300:
297:
295:
292:
291:
290:
289:
286:
283:
282:
277:
274:
272:
271:Related rates
269:
267:
264:
262:
259:
257:
254:
252:
249:
248:
247:
246:
242:
241:
234:
231:
229:
228:of a function
226:
224:
223:infinitesimal
221:
220:
219:
216:
213:
209:
206:
205:
204:
203:
199:
198:
192:
186:
185:
179:
176:
174:
171:
169:
166:
165:
160:
157:
155:
152:
151:
147:
144:
143:
142:
141:
122:
116:
113:
107:
101:
98:
95:
92:
85:
78:
75:
69:
64:
60:
51:
50:
47:
44:
43:
39:
38:
33:
19:
5702:Applications
5660:Crinkled arc
5596:Paley–Wiener
5457:
5294:
5288:
5276:
5243:
5237:
5228:
5209:
5175:q-derivative
5090:
5067:
5053:
5049:finite field
5039:of positive
5037:local fields
5032:
5030:
5023:
5004:
5001:
4986:
4968:
4957:
4934:
4919:
4900:
4704:
4547:
4537:
4514:
4440:
4432:
4428:
4238:
4234:
4224:
4208:
4204:
4202:either way.
4199:binary trees
4184:
4176:
4167:
4132:The mapping
4131:
3897:
3887:
3864:
3844:
3499:difference,
3245:
3229:
3213:
3207:
3202:
3198:
3197:nonzero, if
3194:
3090:q-derivative
2936:
2828:
2694:
2676:
2672:
2670:
2666:Morse theory
2644:matrix, the
2641:
2637:
2633:
2622:
2507:
2345:
1886:
1874:
1868:. See also
1835:
1819:
1810:
1787:vector field
1780:
1764:
1745:
1707:
1685:
1668:
1653:
1644:
1640:
1634:
1627:
1611:vector field
1602:
1598:
1596:
1586:
1581:
1575:
1565:
1560:
1551:
1543:
1538:
1526:
1522:
1518:
1514:
1506:
1502:
1498:
1492:
1441:
1437:
1347:
1340:pathological
1313:
1124:
1123:, is called
1017:
981:
456:Substitution
218:Differential
211:
191:Differential
5468:Holomorphic
5451:Directional
5411:Derivatives
5007:-derivative
4926:subgradient
4911:dimensional
4701:dot product
4445:integration
4187:type theory
3861:Derivations
3855:Lie algebra
3331:time scales
2701:quaternions
2368:, we have
2254:. Then the
1756:Lie algebra
1752:Lie bracket
1613:called the
984:mathematics
912:Precalculus
905:Miscellanea
870:Specialized
777:Definitions
544:Alternating
387:Definitions
200:Definitions
5063:logarithms
4556:divergence
3894:polynomial
3867:derivation
1720:divergence
1592:chain rule
1350:derivative
1188:such that
988:derivative
892:Variations
887:Stochastic
877:Fractional
746:Formalisms
709:Divergence
678:Identities
658:Divergence
208:Derivative
159:Continuity
5591:Regulated
5563:Integrals
5260:0025-584X
4966:instead.
4847:∂
4837:∂
4814:−
4798:∂
4788:∂
4766:∂
4756:∂
4734:∂
4724:∂
4715:◻
4698:Euclidean
4653:∂
4643:∂
4621:∂
4611:∂
4589:∂
4579:∂
4570:Δ
4492:−
4475:−
4412:−
4348:, where
4333:−
4289:−
4271:−
4174:as well.
4143:↦
4101:⋯
4090:−
4074:−
4057:−
4040:−
3975:⋯
3964:−
3948:−
3883:D-modules
3810:−
3787:−
3761:→
3741:−
3718:−
3692:→
3672:−
3652:−
3629:→
3560:−
3557:ω
3531:−
3525:ω
3478:ε
3462:−
3456:ε
3423:−
3400:−
3353:−
3341:ε
3303:−
3267:Δ
3166:−
3143:−
3062:∈
2984:⊂
2956:→
2909:−
2884:−
2848:→
2792:−
2761:−
2743:→
2662:manifolds
2572:α
2559:∈
2521:α
2484:φ
2460:∫
2449:α
2433:−
2415:φ
2409:α
2380:∫
2356:φ
2327:→
2263:α
2231:α
2217:⋯
2206:α
2192:∂
2187:φ
2177:α
2168:∂
2158:φ
2153:α
2122:α
2107:∑
2095:α
2061:α
2054:…
2042:α
2032:α
1966:∞
1953:∈
1950:φ
1930:which is
1913:→
1475:⋅
1455:↦
1397:−
1368:→
1285:‖
1278:‖
1267:‖
1257:−
1242:−
1221:‖
1210:→
1207:‖
1201:‖
1173:→
1138:∈
1068:→
882:Malliavin
769:Geometric
668:Laplacian
618:Dirichlet
529:Geometric
114:−
61:∫
5749:Category
5545:Strongly
5346:Analysis
5100:See also
5084:and the
5013:and the
4997:Jacobian
4560:gradient
4267:′
4253:″
4150:′
4012:′
1797:, and a
1795:manifold
1741:rotation
1712:gradient
1619:gradient
1462:′
1083:, where
1008:geometry
922:Glossary
832:Advanced
810:Jacobian
764:Exterior
694:Gradient
686:Theorems
653:Gradient
592:Integral
554:Binomial
539:Harmonic
404:improper
400:Integral
357:Integral
339:Reynolds
314:Quotient
243:Concepts
79:′
46:Calculus
5711: (
5653:Related
5605:Results
5581:Dunford
5571:Bochner
5537:Bochner
5511:Measure
5268:0030647
5074:locally
4558:of the
4195:algebra
4189:, many
3871:algebra
3001:, then
2664:, see
1862:torsion
1807:subsets
1758:of the
1694:over a
1686:On the
1647:(i.e.,
1010:, etc.
1004:algebra
917:History
815:Hessian
704:Stokes'
699:Green's
531: (
458: (
402: (
324:Inverse
299:Product
210: (
5713:bundle
5541:Weakly
5530:Vector
5301:
5266:
5258:
5220:
4991:, the
4949:module
4905:, the
4519:. The
4211:zipper
2650:tensor
2487:
2481:
2418:
2401:
2000:, and
1856:. In
1846:curves
1840:, the
1729:scalar
1718:, and
1710:, the
1698:, the
1673:, the
1530:matrix
986:, the
759:Tensor
754:Matrix
641:Vector
559:Taylor
517:Series
154:Limits
5434:Total
5201:Notes
5076:like
4427:is a
3892:of a
3853:or a
2082:with
1793:on a
1639:from
1509:is a
582:Ratio
549:Power
468:Euler
446:Discs
441:Parts
309:Power
304:Chain
233:total
5299:ISBN
5256:ISSN
5218:ISBN
5031:The
5002:The
4969:The
4924:and
4920:The
4888:The
4877:The
4684:The
4542:The
3851:ring
3497:Hahn
3251:The
3224:and
3193:For
3088:The
3045:for
1875:The
1820:The
1785:, a
1746:The
1733:flux
1716:curl
1635:For
1628:The
1572:) =J
1018:The
663:Curl
623:Abel
587:Root
5348:in
5248:doi
4987:In
4958:In
4947:or
4935:In
4917:.
4901:In
4703:of
4692:of
4185:In
3754:lim
3685:lim
3622:lim
3255:of
3210:→ 1
2841:lim
2736:lim
2695:In
2640:by
2346:all
1852:or
1836:In
1809:of
1781:In
1735:by
1727:of
1690:of
1669:In
1654:In
1643:to
1601:to
1525:by
1513:on
1501:to
1493:In
1361:lim
1318:of
1197:lim
1127:at
982:In
294:Sum
5751::
5543:/
5539:/
5264:MR
5262:.
5254:.
5242:.
5096:.
5088:.
5061:,
5021:.
4984:.
4939:,
4707::
4531:,
4225:A
4217:.
3865:A
3857:.
3079:.
2687:.
2668:.
2529::=
2267:th
2161::=
2103::=
1828:.
1817:.
1773:.
1714:,
1625:.
1594:.
1584:)J
1574:ƒ(
1490:.
1345:.
1300:0.
1006:,
1002:,
998:,
466:,
462:,
5715:)
5338:e
5331:t
5324:v
5307:.
5270:.
5250::
5244:2
5054:q
5051:F
5005:H
4863:.
4855:2
4851:t
4841:2
4827:2
4823:c
4819:1
4806:2
4802:z
4792:2
4782:+
4774:2
4770:y
4760:2
4750:+
4742:2
4738:x
4728:2
4718:=
4705:R
4669:.
4661:2
4657:z
4647:2
4637:+
4629:2
4625:y
4615:2
4605:+
4597:2
4593:x
4583:2
4573:=
4552:Δ
4548:R
4501:.
4498:)
4495:1
4489:x
4486:4
4483:(
4478:1
4471:L
4467:=
4464:)
4461:x
4458:(
4455:f
4441:x
4433:x
4415:3
4406:x
4403:d
4399:d
4394:2
4391:+
4383:2
4379:x
4375:d
4369:2
4365:d
4359:=
4356:L
4336:1
4330:x
4327:4
4324:=
4321:)
4318:f
4315:(
4312:L
4292:1
4286:x
4283:4
4280:=
4277:f
4274:3
4264:f
4260:2
4257:+
4250:f
4239:x
4237:(
4235:f
4168:R
4147:f
4140:f
4117:.
4112:1
4108:a
4104:+
4098:+
4093:2
4087:d
4083:x
4077:1
4071:d
4067:a
4063:)
4060:1
4054:d
4051:(
4048:+
4043:1
4037:d
4033:x
4027:d
4023:a
4019:d
4016:=
4008:)
4002:0
3998:a
3994:+
3991:x
3986:1
3982:a
3978:+
3972:+
3967:1
3961:d
3957:x
3951:1
3945:d
3941:a
3937:+
3932:d
3928:x
3922:d
3918:a
3913:(
3898:R
3825:.
3819:x
3816:)
3813:1
3807:q
3804:(
3799:)
3796:x
3793:(
3790:f
3784:)
3781:x
3778:q
3775:(
3772:f
3764:1
3758:q
3750:=
3744:x
3738:x
3735:q
3730:)
3727:x
3724:(
3721:f
3715:)
3712:x
3709:q
3706:(
3703:f
3695:1
3689:q
3681:=
3675:x
3669:z
3664:)
3661:x
3658:(
3655:f
3649:)
3646:z
3643:(
3640:f
3632:x
3626:z
3601:x
3598:q
3595:=
3592:z
3569:.
3563:x
3554:+
3551:x
3548:q
3543:)
3540:x
3537:(
3534:f
3528:)
3522:+
3519:x
3516:q
3513:(
3510:f
3483:.
3474:)
3471:x
3468:(
3465:f
3459:)
3453:+
3450:x
3447:(
3444:f
3438:=
3432:x
3429:)
3426:1
3420:q
3417:(
3412:)
3409:x
3406:(
3403:f
3397:)
3394:x
3391:q
3388:(
3385:f
3362:x
3359:)
3356:1
3350:q
3347:(
3344:=
3315:)
3312:x
3309:(
3306:f
3300:)
3297:1
3294:+
3291:x
3288:(
3285:f
3282:=
3279:)
3276:x
3273:(
3270:f
3246:q
3230:q
3214:q
3208:q
3203:x
3199:f
3195:x
3181:.
3175:x
3172:)
3169:1
3163:q
3160:(
3155:)
3152:x
3149:(
3146:f
3140:)
3137:x
3134:q
3131:(
3128:f
3122:=
3119:)
3116:x
3113:(
3110:f
3105:q
3101:D
3066:H
3059:b
3056:,
3053:a
3033:b
3030:q
3027:+
3024:a
3021:=
3018:)
3015:q
3012:(
3009:f
2988:H
2981:U
2960:H
2952:H
2948::
2945:f
2922:.
2918:]
2912:1
2905:h
2900:)
2896:)
2893:a
2890:(
2887:f
2881:)
2878:h
2875:+
2872:a
2869:(
2866:f
2862:(
2857:[
2851:0
2845:h
2813:]
2808:)
2804:)
2801:a
2798:(
2795:f
2789:)
2786:h
2783:+
2780:a
2777:(
2774:f
2770:(
2764:1
2757:h
2752:[
2746:0
2740:h
2711:H
2677:n
2673:n
2642:n
2638:n
2634:n
2598:)
2593:n
2588:R
2583:(
2576:|
2568:|
2563:C
2556:u
2532:v
2526:u
2517:D
2493:x
2490:d
2478:v
2471:n
2466:R
2453:|
2445:|
2440:)
2436:1
2430:(
2427:=
2424:x
2421:d
2405:D
2398:u
2391:n
2386:R
2331:R
2322:n
2317:R
2312::
2309:v
2289:u
2235:n
2225:n
2221:x
2210:1
2200:1
2196:x
2181:|
2173:|
2149:D
2126:i
2116:n
2111:1
2099:|
2091:|
2070:)
2065:n
2057:,
2051:,
2046:1
2038:(
2035:=
2012:n
1987:)
1982:n
1977:R
1972:(
1961:c
1957:C
1917:R
1908:n
1903:R
1898::
1895:u
1811:R
1708:R
1645:R
1641:R
1605:(
1603:R
1599:R
1587:x
1582:g
1580:(
1578:)
1576:x
1570:f
1568:°
1566:g
1564:(
1561:x
1556:f
1554:°
1552:g
1544:x
1539:x
1536:J
1527:n
1523:m
1515:R
1507:A
1503:R
1499:R
1478:t
1472:)
1469:x
1466:(
1459:f
1452:t
1442:A
1438:A
1424:,
1421:A
1418:=
1413:h
1409:)
1406:x
1403:(
1400:f
1394:)
1391:h
1388:+
1385:x
1382:(
1379:f
1371:0
1365:h
1326:x
1297:=
1289:V
1281:h
1271:W
1263:h
1260:A
1254:)
1251:x
1248:(
1245:f
1239:)
1236:h
1233:+
1230:x
1227:(
1224:f
1213:0
1204:h
1176:W
1170:V
1167::
1164:A
1141:U
1135:x
1111:V
1091:U
1071:W
1065:U
1062::
1059:f
1039:W
1036:,
1033:V
971:e
964:t
957:v
535:)
470:)
406:)
214:)
126:)
123:a
120:(
117:f
111:)
108:b
105:(
102:f
99:=
96:t
93:d
89:)
86:t
83:(
76:f
70:b
65:a
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.