4200:
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.
4867:
1645:), 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.
5085:
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
4195:
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
4199:
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
4704:
4673:
3829:
1671:
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.
4121:
2246:
2497:
2617:
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
3487:
4559:
3610:
1716:
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
1304:
2642:. 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
1992:
4886:
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.
2603:
4862:{\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}}}.}
4419:
3573:
1428:
2074:
3185:
2137:
4433:
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
125:
3900:
2368:
2336:
1922:
2132:
4296:
2965:
1482:
3319:
2993:
2926:
3371:
3366:
3071:
2818:
5412:
5330:
4505:
4155:
2273:
2536:
2716:
4340:
3037:
2360:
1180:
1075:
2609:, the dual space of test functions. Weak derivatives are particularly useful in the study of partial differential equations, and within parts of functional analysis.
1185:
1145:
5087:
3605:
1043:
2293:
2016:
1330:
1115:
1095:
4903:
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
5059:
and others the derivative can be used to develop notions of smoothness, analycity, integration, Taylor series as well as a theory of differential equations.
1544:, 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
4974:. Fréchet differentiability is a strictly stronger condition than Gateaux differentiability, even in finite dimensions. Between the two extremes is the
5323:
4171:
The notion of derivation applies to noncommutative as well as commutative rings, and even to non-associative algebraic structures, such as Lie algebras.
5210:, Garrett Sobczyk: Clifford Algebra to Geometric Calculus, a Unified Language for mathematics and Physics (Dordrecht/Boston:G.Reidel Publ.Co., 1984,
1763:(a degree -1 derivation on the exterior algebra defined by contraction with a vector field), the exterior derivative and the Lie derivative form a
5062:
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
5051:(it is known that any local field of positive characteristic is isomorphic to a Laurent series field). Along with suitably defined analogs to the
4345:
3226:-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
3496:
4668:{\displaystyle \Delta ={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}.}
1626:
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.
5316:
1349:
1939:
3089:
1725:
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
4223:
combines several derivatives, possibly of different orders, in one algebraic expression. This is especially useful in considering ordinary
3824:{\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}}.}
221:
5748:
2545:
1744:
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
5637:
5492:
4993:, used for changing variables, to measures. It expresses one measure μ in terms of another measure ν (under certain conditions).
2241:{\textstyle D^{\alpha }\varphi :={\frac {\partial ^{|\alpha |}\varphi }{\partial x_{1}^{\alpha _{1}}\dotsm x_{n}^{\alpha _{n}}}}}
2021:
5627:
5296:
4116:{\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}.}
3256:
2492:{\displaystyle \int _{\mathbb {R} ^{n}}u\ D^{\alpha }\!\varphi \ dx=(-1)^{|\alpha |}\int _{\mathbb {R} ^{n}}v\ \varphi \ dx}
5384:
487:
462:
5417:
4682:
or wave operator is similar to the
Laplacian, but acts on functions of four variables. Its definition uses the indefinite
4444:
2298:
1884:
3323:
The q-derivative, the difference operator and the standard derivative can all be viewed as the same thing on different
963:
526:
44:
5215:
4945:. They can be used to define an analogue of exterior derivative from differential geometry that applies to arbitrary
3575:
The Hahn difference is not only a generalization of the q-derivative but also an extension of the forward difference.
2079:
482:
4191:
generated by a transformation that maps structures based on the type back into the type. For example, the type T of
2934:
467:
5029:
is an operation similar to usual differentiation but with the usual context of real or complex numbers changed to
5686:
5461:
4990:
4208:
3868:
1527:
803:
477:
452:
134:
5681:
5513:
3879:, but also turn up in many other areas, where they often agree with less algebraic definitions of derivatives.
2626:
with respect to different variables. For example, the second order partial derivatives of a scalar function of
2718:
are not commutative, the limit of the difference quotient yields two different derivatives: A left derivative
4510:
3860:
1870:
585:
532:
413:
5538:
4986:
4526:
4224:
2970:
2651:
1441:
239:
211:
5622:
2829:
322:
5530:
5141:
4879:, in much the same way that a normal derivative describes how a function is approximated by a linear map.
3330:
3042:
2724:
1925:
1309:
836:
444:
282:
254:
20:
5605:
5534:
3872:
2606:
1863:
1815:
1668:
1333:
707:
671:
448:
327:
216:
206:
2673:, there are various ways to define derivatives of fractional or negative orders, which are studied in
1818:
of a map between manifolds is the induced map between tangent spaces of those maps. It abstracts the
5676:
5589:
5518:
5232:
5156:
5105:
4238:
2505:
2251:
1936:. First define test functions, which are infinitely differentiable and compactly supported functions
471:
307:
5610:
5497:
5343:
5034:
4971:
606:
166:
4875:
is a non-linear differential operator which describes how a complex function is approximated by a
2699:
5712:
4957:
4204:
4129:
1630:
1148:
920:
712:
601:
5615:
5444:
5362:
5188:
5174:
5147:
5135:
5086:
classical differential operator has a discrete analog, the difference operator, there are also
4908:
4301:
3578:
Also note that the q-derivative is nothing but a special case of the familiar derivative. Take
2998:
2619:
1859:
1808:
1796:
1718:
1488:
956:
885:
846:
730:
666:
590:
4934:
2345:
1153:
1048:
5504:
5434:
5357:
5339:
5111:
5099:
5075:
5038:
5004:
4900:
4883:
4876:
4872:
4220:
2690:
1831:
1776:
1623:
989:
985:
930:
596:
367:
312:
273:
179:
5422:
4517:
which assigns to each function its derivative is an example of a differential operator on a
3482:{\displaystyle {\frac {f(qx)-f(x)}{(q-1)x}}={\frac {f(x+\varepsilon )-f(x)}{\varepsilon }}.}
1124:
1013:
5632:
5546:
5487:
5379:
5261:
5179:
5162:
5079:
5052:
5019:
5012:
3883:
3864:
3250:
1995:
1933:
1835:
1795:
may be defined as a derivation at a point. This allows the abstraction of the notion of a
1753:
1722:
1653:
1523:
1515:
of a function. If such an operator exists, then it is unique, and can be represented by an
935:
915:
841:
510:
429:
403:
317:
3581:
8:
5668:
5658:
5541:
5456:
4946:
4930:
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3324:
3246:
2674:
1851:
1697:
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1616:
1022:
1017:
910:
880:
870:
757:
611:
408:
264:
147:
142:
5722:
5308:
5274:
5022:. There is no completely satisfactory analog of the first-order derivative or gradient.
5584:
5439:
5123:
5008:
4964:
4430:
4184:
3844:
2693:, derivatives can be defined in a similar way to real and complex functions. Since the
2623:
2278:
2001:
1730:
1709:
1685:
1664:
1541:
1315:
1100:
1080:
875:
778:
762:
702:
697:
692:
656:
537:
456:
362:
357:
161:
156:
2622:. In this case, instead of repeatedly applying the derivative, one repeatedly applies
5292:
5249:
5211:
4522:
4188:
4165:
3227:
2647:
2539:
949:
783:
561:
439:
392:
249:
244:
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5241:
5129:
5063:
4975:
4942:
4938:
4679:
4537:
3219:
3215:
1847:
1764:
1760:
1681:
1649:
1642:
1608:
793:
687:
661:
522:
434:
398:
4956:, the usual definition of derivative is not quite strong enough, and one requires
5702:
5579:
5569:
5394:
5389:
5257:
5117:
4953:
4923:
4691:
4687:
4514:
4158:
1929:
1819:
1784:
1689:
1657:
1504:
1336:
925:
798:
752:
747:
634:
547:
492:
4235:) is a twice differentiable function of one variable, the differential equation
1299:{\displaystyle \lim _{\|h\|\to 0}{\frac {\|f(x+h)-f(x)-Ah\|_{W}}{\|h\|_{V}}}=0.}
5717:
5523:
5207:
4982:
4968:
4518:
4203:
This concept of a derivative of a type has practical applications, such as the
2931:
The existence of these limits are very restrictive conditions. For example, if
2678:
2646:). Nevertheless, higher derivatives have important applications to analysis of
2639:
1855:
1792:
1741:
1491:, in the context of differential equations defined by a vector valued function
808:
616:
383:
3839:
In algebra, generalizations of the derivative can be obtained by imposing the
1842:. This extends the directional derivative of scalar functions to sections of
5742:
5706:
5482:
5374:
5369:
5253:
5245:
5227:
4915:
4683:
3490:
3235:
3231:
3211:
2677:. The −1 order derivative corresponds to the integral, whence the term
1843:
1700:
and squares to zero. It is a grade 1 derivation on the exterior algebra. In
993:
788:
552:
302:
259:
4675:
Analogous operators can be defined for functions of any number of variables.
3867:
which satisfies the
Leibniz law (the product rule). Higher derivatives and
5653:
5508:
5168:
5071:
5042:
5030:
4997:
3840:
3083:
2659:
1780:
1604:
1600:
542:
287:
19:
This article is about the term as used in mathematics. For other uses, see
1332:, rather than at individual points, as not doing so tends to lead to many
5230:(1949). "Über Orthogonalpolynome, die q-Differenzengleichungen genügen".
4919:
4694:
4532:
Some of these operators are so important that they have their own names:
4192:
4180:
3848:
1838:
makes a choice for taking directional derivatives of vector fields along
1749:
1745:
977:
905:
2542:. This definition coincides with the classical derivative for functions
1987:{\displaystyle \varphi \in C_{c}^{\infty }\left(\mathbb {R} ^{n}\right)}
5404:
5056:
4549:
3887:
3871:
can also be defined. They are studied in a purely algebraic setting in
3214:. A large body of results from ordinary differential calculus, such as
2694:
1713:
1585:
1343:
981:
651:
575:
297:
292:
196:
5159: – Use of numerical analysis to estimate derivatives of functions
4509:
Combining derivatives of different variables results in a notion of a
4907:
vector space. An important case is the variational derivative in the
4904:
580:
570:
5556:
5074:. Thus one might want a derivative with some of the features of a
4553:
4438:
3876:
3076:
2655:
1799:
of a scalar function to general manifolds. For manifolds that are
1788:
1734:
1705:
1612:
1001:
646:
388:
345:
34:
5067:
5018:
Laplacians and differential equations using the
Laplacian can be
1511:
considered as a vector space over itself, and corresponds to the
997:
5114: – Function defined on formal languages in computer science
2605:, and can be extended to a type of generalized functions called
3238:
have changed the situation dramatically, and the popularity of
2643:
2598:{\displaystyle u\in C^{|\alpha |}\left(\mathbb {R} ^{n}\right)}
1800:
1342:
The Fréchet derivative is quite similar to the formula for the
4423:
second order linear constant coefficient differential operator
2654:. For an advanced application of this analysis to topology of
1756:
of the manifold). It is a grade 0 derivation on the algebra.
988:
and admits many possible generalizations within the fields of
2967:
has left-derivatives at every point on an open connected set
1839:
4414:{\displaystyle L={\frac {d^{2}}{dx^{2}}}+2{\frac {d}{dx}}-3}
5413:
Differentiable vector–valued functions from
Euclidean space
3568:{\displaystyle {\frac {f(qx+\omega )-f(x)}{qx+\omega -x}}.}
1726:
1308:
Functions are defined as being differentiable in some open
4556:
of a scalar function of three variables, or explicitly as
2612:
5338:
5108: – Numerical calculations carrying along derivatives
1675:
1423:{\displaystyle \lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}=A,}
2069:{\displaystyle \alpha =(\alpha _{1},\dots ,\alpha _{n})}
1873:
extends the exterior derivative to vector valued forms.
5152:
Pages displaying short descriptions of redirect targets
3253:
is another discrete analog of the standard derivative.
3180:{\displaystyle D_{q}f(x)={\frac {f(qx)-f(x)}{(q-1)x}}.}
1854:, the existence of a metric chooses a unique preferred
1434:
to the left hand side. However, the Fréchet derivative
5277:, Robert S. Strichartz - Article in Notices of the AMS
2254:
2140:
2082:
1552:
is a product of corresponding
Jacobian matrices: J
5102: – Function defined on integers in number theory
4707:
4562:
4447:
4429:. The key idea here is that we consider a particular
4348:
4304:
4241:
4132:
3903:
3613:
3584:
3499:
3374:
3333:
3259:
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2937:
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2727:
2702:
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2371:
2348:
2301:
2281:
2024:
2004:
1942:
1887:
1444:
1352:
1318:
1188:
1156:
1127:
1103:
1083:
1051:
1025:
47:
5184:
Pages displaying wikidata descriptions as a fallback
5088:
discrete analogs of these multiplicative derivatives
4529:
can be defined which allow for fractional calculus.
1928:, but not necessarily classically differentiable, a
1584:(ƒ). This is a higher-dimensional statement of the
5120: – Class of generalisations of the derivative
4861:
4667:
4499:
4413:
4334:
4290:
4149:
4115:
3823:
3599:
3567:
3481:
3360:
3313:
3179:
3065:
3031:
2987:
2959:
2920:
2812:
2710:
2597:
2530:
2491:
2354:
2331:{\displaystyle v:\mathbb {R} ^{n}\to \mathbb {R} }
2330:
2287:
2267:
2240:
2126:
2068:
2010:
1986:
1917:{\displaystyle u:\mathbb {R} ^{n}\to \mathbb {R} }
1916:
1476:
1422:
1324:
1298:
1174:
1139:
1109:
1089:
1069:
1037:
119:
2407:
16:Fundamental construction of differential calculus
5740:
5126: – Generalization of derivative to fractals
4544:is a second-order partial differential operator
3747:
3678:
3615:
3077:Difference operator, q-analogues and time scales
2834:
2729:
2127:{\textstyle |\alpha |:=\sum _{1}^{n}\alpha _{i}}
1354:
1190:
120:{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}
5165: – Type of derivative of a linear operator
5144: – Method of mathematical differentiation
2960:{\displaystyle f:\mathbb {H} \to \mathbb {H} }
1783:may be defined as a derivation on the ring of
5324:
2669: th derivatives for any natural number
957:
4227:with constant coefficients. For example, if
3206:we obtain the ordinary derivative, thus the
1656:, which are complex-valued functions on the
1278:
1271:
1260:
1214:
1200:
1194:
5171: – Q-analog of the ordinary derivative
2684:
1696:is the unique linear map which satisfies a
1603:), the Fréchet derivative corresponds to a
1346:found in elementary one-variable calculus,
5331:
5317:
5138: – Mathematical operation in calculus
5003:is a notion of derivative in the study of
3489:The q-derivative is a special case of the
1807:, this tangent vector will agree with the
964:
950:
4922:are generalizations of the derivative to
4437: − 1, by analogy with ordinary
4214:
3834:
3059:
2981:
2953:
2945:
2704:
2581:
2459:
2379:
2324:
2310:
1970:
1910:
1896:
1858:-free covariant derivative, known as the
1540:. Each entry of this matrix represents a
80:
5628:No infinite-dimensional Lebesgue measure
5291:. New York: Cambridge University Press.
5286:
5182: – generalization of the derivative
4890:
3086:of a function is defined by the formula
1770:
5638:Structure theorem for Gaussian measures
4174:
3314:{\displaystyle \Delta f(x)=f(x+1)-f(x)}
2613:Higher-order and fractional derivatives
1825:
488:Differentiating under the integral sign
5741:
4164:. This definition can be extended to
1676:Exterior derivative and Lie derivative
5514:infinite-dimensional Gaussian measure
5312:
3843:in an algebraic structure, such as a
2988:{\displaystyle U\subset \mathbb {H} }
1866:for a treatment oriented to physics.
1660:where the Fréchet derivative exists.
1477:{\displaystyle t\mapsto f'(x)\cdot t}
1007:
5385:Infinite-dimensional vector function
5226:
4949:, instead of just smooth manifolds.
2921:{\displaystyle \lim _{h\to 0}\left.}
5289:Analysis in Positive Characteristic
3361:{\displaystyle \varepsilon =(q-1)x}
3066:{\displaystyle a,b\in \mathbb {H} }
2813:{\displaystyle \lim _{h\to 0}\left}
2630:variables can be organized into an
1876:
1737:" a vector field has near a point.
1652:, the central objects of study are
1016:defines the derivative for general
13:
4967:extends the Fréchet derivative to
4840:
4830:
4791:
4781:
4759:
4749:
4727:
4717:
4646:
4636:
4614:
4604:
4582:
4572:
4563:
4500:{\displaystyle f(x)=L^{-1}(4x-1).}
3260:
2185:
2161:
1959:
1615:but it is more natural to use the
29:Part of a series of articles about
14:
5760:
5749:Generalizations of the derivative
5452:Generalizations of the derivative
5418:Differentiation in Fréchet spaces
3210:-derivative may be viewed as its
1698:graded version of the Leibniz law
1611:. This can be interpreted as the
984:is a fundamental construction of
5066:, one studies spaces which look
4209:functional programming languages
3869:algebraic differential operators
3195:is a differentiable function of
2502:If such a function exists, then
2268:{\textstyle \alpha ^{\text{th}}}
5687:Holomorphic functional calculus
4937:are universal derivations of a
4291:{\displaystyle f''+2f'-3f=4x-1}
3841:Leibniz rule of differentiation
2531:{\displaystyle D^{\alpha }u:=v}
1591:For real valued functions from
5682:Continuous functional calculus
5280:
5268:
5220:
5201:
5011:. It is used in the study of
4491:
4476:
4457:
4451:
4314:
4308:
4298:may be rewritten in the form
4136:
4056:
4044:
3863:is a linear map on a ring or
3854:
3809:
3797:
3792:
3786:
3777:
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3754:
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3536:
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2423:
2320:
2295:exists if there is a function
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2084:
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2031:
1906:
1536:(ƒ) of the mapping ƒ at point
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1:
5287:Kochubei, Anatoly N. (2009).
5150: – Branch of mathematics
4527:pseudo-differential operators
4511:partial differential operator
4225:linear differential equations
2134:. Applied to test functions,
1871:exterior covariant derivative
414:Integral of inverse functions
5132: – Mathematical concept
4157:is then a derivation on the
2711:{\displaystyle \mathbb {H} }
7:
5142:Logarithmic differentiation
5093:
4150:{\displaystyle f\mapsto f'}
1932:may be defined by means of
1816:differential or pushforward
837:Calculus on Euclidean space
255:Logarithmic differentiation
21:derivative (disambiguation)
10:
5765:
5041:with coefficients in some
3873:differential Galois theory
3242:-analogues is on the rise.
1864:gauge covariant derivative
1733:. Curl measures how much "
18:
5695:
5677:Borel functional calculus
5667:
5646:
5598:
5555:
5475:
5403:
5350:
5344:topological vector spaces
5233:Mathematische Nachrichten
5157:Numerical differentiation
5106:Automatic differentiation
4972:topological vector spaces
4926:used in convex analysis.
4335:{\displaystyle L(f)=4x-1}
3032:{\displaystyle f(q)=a+qb}
1513:best linear approximation
1499:, the Fréchet derivative
571:Summand limit (term test)
5611:Inverse function theorem
5498:Classical Wiener measure
5246:10.1002/mana.19490020103
5194:
4987:Radon–Nikodym derivative
4958:strict differentiability
4187:can be described as the
3890:over a commutative ring
2685:Quaternionic derivatives
2355:{\displaystyle \varphi }
1748:(vector fields form the
1175:{\displaystyle A:V\to W}
1070:{\displaystyle f:U\to W}
250:Implicit differentiation
240:Differentiation notation
167:Inverse function theorem
5713:Convenient vector space
4425:acting on functions of
2823:and a right derivative
1719:directional derivatives
1631:vector-valued functions
1149:bounded linear operator
713:Helmholtz decomposition
5606:Cameron–Martin theorem
5363:Classical Wiener space
5189:Topological derivative
5175:Semi-differentiability
5148:Non-classical analysis
5136:Logarithmic derivative
5005:abstract Wiener spaces
4909:calculus of variations
4863:
4669:
4501:
4441:, and formally write
4415:
4336:
4292:
4215:Differential operators
4151:
4117:
3835:Derivatives in algebra
3825:
3601:
3569:
3483:
3362:
3327:. For example, taking
3315:
3181:
3067:
3033:
2989:
2961:
2922:
2814:
2712:
2620:multivariable calculus
2599:
2532:
2493:
2356:
2332:
2289:
2269:
2242:
2128:
2113:
2070:
2012:
1988:
1918:
1860:Levi-Civita connection
1809:directional derivative
1797:directional derivative
1489:multivariable calculus
1478:
1424:
1326:
1300:
1176:
1141:
1140:{\displaystyle x\in U}
1119:Fréchet differentiable
1111:
1091:
1071:
1045:. Briefly, a function
1039:
847:Limit of distributions
667:Directional derivative
323:Faà di Bruno's formula
121:
5623:Feldman–Hájek theorem
5435:Functional derivative
5358:Abstract Wiener space
5112:Brzozowski derivative
5100:Arithmetic derivative
5076:functional derivative
5039:formal Laurent series
4901:functional derivative
4891:Other generalizations
4884:Wirtinger derivatives
4877:fractional-linear map
4873:Schwarzian derivative
4864:
4670:
4502:
4416:
4337:
4293:
4221:differential operator
4152:
4118:
3826:
3602:
3570:
3484:
3363:
3316:
3234:and the discovery of
3199:then in the limit as
3182:
3068:
3034:
2990:
2962:
2923:
2815:
2713:
2691:quaternionic analysis
2650:of a function at its
2600:
2533:
2494:
2357:
2333:
2290:
2270:
2243:
2129:
2099:
2071:
2013:
1989:
1919:
1832:differential geometry
1777:differential topology
1771:Differential topology
1654:holomorphic functions
1624:convective derivative
1479:
1438:denotes the function
1425:
1327:
1301:
1177:
1142:
1112:
1097:is an open subset of
1092:
1072:
1040:
990:mathematical analysis
986:differential calculus
931:Mathematical analysis
842:Generalized functions
527:arithmetico-geometric
368:Leibniz integral rule
122:
5547:Radonifying function
5488:Cylinder set measure
5380:Cylinder set measure
5275:Analysis on Fractals
5180:Symmetric derivative
5163:Pincherle derivative
5080:covariant derivative
5053:exponential function
5013:stochastic processes
4935:Kähler differentials
4705:
4560:
4445:
4346:
4302:
4239:
4175:Derivative of a type
4130:
3901:
3611:
3600:{\displaystyle z=qx}
3582:
3497:
3372:
3331:
3257:
3251:difference equations
3090:
3043:
2999:
2971:
2935:
2830:
2725:
2700:
2546:
2506:
2369:
2346:
2299:
2279:
2252:
2138:
2080:
2022:
2002:
1940:
1934:integration by parts
1885:
1836:covariant derivative
1826:Covariant derivative
1754:diffeomorphism group
1669:geometric derivative
1442:
1350:
1316:
1186:
1154:
1125:
1101:
1081:
1049:
1023:
1018:normed vector spaces
936:Nonstandard analysis
404:Lebesgue integration
274:Rules and identities
45:
5669:Functional calculus
5659:Covariance operator
5580:Gelfand–Pettis/Weak
5542:measurable function
5457:Hadamard derivative
5020:defined on fractals
4947:algebraic varieties
4931:commutative algebra
4897:functional analysis
4185:abstract data types
3247:difference operator
2675:fractional calculus
2624:partial derivatives
2275:weak derivative of
2234:
2209:
1998:, which are length
1963:
1852:Riemannian geometry
1694:exterior derivative
1617:exterior derivative
1038:{\displaystyle V,W}
607:Cauchy condensation
409:Contour integration
135:Fundamental theorem
62:
5616:Nash–Moser theorem
5493:Canonical Gaussian
5440:Gateaux derivative
5423:Fréchet derivative
5124:Fractal derivative
5027:Carlitz derivative
5009:Malliavin calculus
4965:Gateaux derivative
4859:
4665:
4521:. By means of the
4497:
4431:linear combination
4411:
4332:
4288:
4207:technique used in
4166:rational functions
4147:
4113:
3875:and the theory of
3821:
3761:
3692:
3629:
3597:
3565:
3479:
3358:
3311:
3230:. The progress of
3177:
3063:
3029:
2985:
2957:
2918:
2848:
2810:
2743:
2708:
2595:
2538:, which is unique
2528:
2489:
2352:
2328:
2285:
2265:
2238:
2213:
2188:
2124:
2066:
2018:lists of integers
2008:
1984:
1949:
1926:locally integrable
1914:
1759:Together with the
1731:divergence theorem
1686:differential forms
1665:geometric calculus
1542:partial derivative
1474:
1420:
1368:
1322:
1296:
1210:
1172:
1147:if there exists a
1137:
1107:
1087:
1067:
1035:
1014:Fréchet derivative
1008:Fréchet derivative
779:Partial derivative
708:generalized Stokes
602:Alternating series
483:Reduction formulae
472:Heaviside's method
453:tangent half-angle
440:Cylindrical shells
363:Integral transform
358:Lists of integrals
162:Mean value theorem
117:
48:
5736:
5735:
5633:Sazonov's theorem
5519:Projection-valued
5298:978-0-521-50977-0
4854:
4825:
4805:
4773:
4741:
4690:, instead of the
4660:
4628:
4596:
4523:Fourier transform
4403:
4382:
3884:formal derivative
3882:For example, the
3816:
3746:
3741:
3677:
3672:
3614:
3560:
3474:
3429:
3228:special functions
3172:
2833:
2728:
2540:almost everywhere
2482:
2476:
2413:
2396:
2288:{\displaystyle u}
2262:
2236:
2011:{\displaystyle n}
1881:Given a function
1848:principal bundles
1643:parametric curves
1430:and simply moves
1409:
1353:
1325:{\displaystyle x}
1288:
1189:
1110:{\displaystyle V}
1090:{\displaystyle U}
974:
973:
854:
853:
816:
815:
784:Multiple integral
720:
719:
624:
623:
591:Direct comparison
562:Convergence tests
500:
499:
468:Partial fractions
335:
334:
245:Second derivative
5756:
5728:Hilbert manifold
5723:Fréchet manifold
5507: like
5467:Quasi-derivative
5333:
5326:
5319:
5310:
5309:
5303:
5302:
5284:
5278:
5272:
5266:
5265:
5224:
5218:
5205:
5185:
5153:
5130:Hasse derivative
5064:Finsler geometry
4989:generalizes the
4976:quasi-derivative
4939:commutative ring
4924:convex functions
4868:
4866:
4865:
4860:
4855:
4853:
4852:
4851:
4838:
4837:
4828:
4826:
4824:
4823:
4811:
4806:
4804:
4803:
4802:
4789:
4788:
4779:
4774:
4772:
4771:
4770:
4757:
4756:
4747:
4742:
4740:
4739:
4738:
4725:
4724:
4715:
4674:
4672:
4671:
4666:
4661:
4659:
4658:
4657:
4644:
4643:
4634:
4629:
4627:
4626:
4625:
4612:
4611:
4602:
4597:
4595:
4594:
4593:
4580:
4579:
4570:
4547:
4540:or Laplacian on
4538:Laplace operator
4506:
4504:
4503:
4498:
4475:
4474:
4420:
4418:
4417:
4412:
4404:
4402:
4391:
4383:
4381:
4380:
4379:
4366:
4365:
4356:
4341:
4339:
4338:
4333:
4297:
4295:
4294:
4289:
4263:
4249:
4156:
4154:
4153:
4148:
4146:
4122:
4120:
4119:
4114:
4109:
4108:
4090:
4089:
4074:
4073:
4040:
4039:
4024:
4023:
4008:
4004:
4000:
3999:
3998:
3983:
3982:
3964:
3963:
3948:
3947:
3929:
3928:
3919:
3918:
3830:
3828:
3827:
3822:
3817:
3815:
3795:
3763:
3760:
3742:
3740:
3726:
3694:
3691:
3673:
3671:
3660:
3631:
3628:
3607:. Then we have,
3606:
3604:
3603:
3598:
3574:
3572:
3571:
3566:
3561:
3559:
3539:
3501:
3488:
3486:
3485:
3480:
3475:
3470:
3435:
3430:
3428:
3408:
3376:
3367:
3365:
3364:
3359:
3320:
3318:
3317:
3312:
3220:Taylor expansion
3216:binomial formula
3205:
3186:
3184:
3183:
3178:
3173:
3171:
3151:
3119:
3102:
3101:
3072:
3070:
3069:
3064:
3062:
3038:
3036:
3035:
3030:
2994:
2992:
2991:
2986:
2984:
2966:
2964:
2963:
2958:
2956:
2948:
2927:
2925:
2924:
2919:
2914:
2910:
2909:
2908:
2896:
2892:
2847:
2819:
2817:
2816:
2811:
2809:
2805:
2804:
2800:
2761:
2760:
2742:
2717:
2715:
2714:
2709:
2707:
2604:
2602:
2601:
2596:
2594:
2590:
2589:
2584:
2574:
2573:
2572:
2564:
2537:
2535:
2534:
2529:
2518:
2517:
2498:
2496:
2495:
2490:
2480:
2474:
2470:
2469:
2468:
2467:
2462:
2451:
2450:
2449:
2441:
2411:
2406:
2405:
2394:
2390:
2389:
2388:
2387:
2382:
2361:
2359:
2358:
2353:
2337:
2335:
2334:
2329:
2327:
2319:
2318:
2313:
2294:
2292:
2291:
2286:
2274:
2272:
2271:
2266:
2264:
2263:
2260:
2247:
2245:
2244:
2239:
2237:
2235:
2233:
2232:
2231:
2221:
2208:
2207:
2206:
2196:
2183:
2179:
2178:
2177:
2169:
2158:
2150:
2149:
2133:
2131:
2130:
2125:
2123:
2122:
2112:
2107:
2095:
2087:
2075:
2073:
2072:
2067:
2062:
2061:
2043:
2042:
2017:
2015:
2014:
2009:
1993:
1991:
1990:
1985:
1983:
1979:
1978:
1973:
1962:
1957:
1923:
1921:
1920:
1915:
1913:
1905:
1904:
1899:
1877:Weak derivatives
1785:smooth functions
1765:Lie superalgebra
1761:interior product
1682:exterior algebra
1650:complex analysis
1609:total derivative
1483:
1481:
1480:
1475:
1458:
1429:
1427:
1426:
1421:
1410:
1405:
1370:
1367:
1331:
1329:
1328:
1323:
1305:
1303:
1302:
1297:
1289:
1287:
1286:
1285:
1269:
1268:
1267:
1212:
1209:
1181:
1179:
1178:
1173:
1146:
1144:
1143:
1138:
1116:
1114:
1113:
1108:
1096:
1094:
1093:
1088:
1076:
1074:
1073:
1068:
1044:
1042:
1041:
1036:
966:
959:
952:
900:
865:
831:
830:
827:
794:Surface integral
737:
736:
733:
641:
640:
637:
597:Limit comparison
517:
516:
513:
399:Riemann integral
352:
351:
348:
308:L'Hôpital's rule
265:Taylor's theorem
186:
185:
182:
126:
124:
123:
118:
70:
61:
56:
26:
25:
5764:
5763:
5759:
5758:
5757:
5755:
5754:
5753:
5739:
5738:
5737:
5732:
5703:Banach manifold
5691:
5663:
5642:
5594:
5570:Direct integral
5551:
5471:
5399:
5395:Vector calculus
5390:Matrix calculus
5346:
5337:
5307:
5306:
5299:
5285:
5281:
5273:
5269:
5225:
5221:
5206:
5202:
5197:
5183:
5151:
5118:Dini derivative
5096:
5050:
5037:in the form of
4954:p-adic analysis
4893:
4847:
4843:
4839:
4833:
4829:
4827:
4819:
4815:
4810:
4798:
4794:
4790:
4784:
4780:
4778:
4766:
4762:
4758:
4752:
4748:
4746:
4734:
4730:
4726:
4720:
4716:
4714:
4706:
4703:
4702:
4688:Minkowski space
4653:
4649:
4645:
4639:
4635:
4633:
4621:
4617:
4613:
4607:
4603:
4601:
4589:
4585:
4581:
4575:
4571:
4569:
4561:
4558:
4557:
4545:
4515:linear operator
4467:
4463:
4446:
4443:
4442:
4395:
4390:
4375:
4371:
4367:
4361:
4357:
4355:
4347:
4344:
4343:
4303:
4300:
4299:
4256:
4242:
4240:
4237:
4236:
4217:
4177:
4159:polynomial ring
4139:
4131:
4128:
4127:
4104:
4100:
4079:
4075:
4063:
4059:
4029:
4025:
4019:
4015:
3994:
3990:
3978:
3974:
3953:
3949:
3937:
3933:
3924:
3920:
3914:
3910:
3909:
3905:
3904:
3902:
3899:
3898:
3857:
3837:
3796:
3764:
3762:
3750:
3727:
3695:
3693:
3681:
3661:
3632:
3630:
3618:
3612:
3609:
3608:
3583:
3580:
3579:
3540:
3502:
3500:
3498:
3495:
3494:
3436:
3434:
3409:
3377:
3375:
3373:
3370:
3369:
3332:
3329:
3328:
3258:
3255:
3254:
3222:, have natural
3200:
3152:
3120:
3118:
3097:
3093:
3091:
3088:
3087:
3079:
3058:
3044:
3041:
3040:
3000:
2997:
2996:
2980:
2972:
2969:
2968:
2952:
2944:
2936:
2933:
2932:
2901:
2897:
2858:
2854:
2853:
2849:
2837:
2831:
2828:
2827:
2766:
2762:
2753:
2749:
2748:
2744:
2732:
2726:
2723:
2722:
2703:
2701:
2698:
2697:
2687:
2665:In addition to
2652:critical points
2615:
2585:
2580:
2579:
2575:
2568:
2560:
2559:
2555:
2547:
2544:
2543:
2513:
2509:
2507:
2504:
2503:
2463:
2458:
2457:
2456:
2452:
2445:
2437:
2436:
2432:
2401:
2397:
2383:
2378:
2377:
2376:
2372:
2370:
2367:
2366:
2347:
2344:
2343:
2342:test functions
2323:
2314:
2309:
2308:
2300:
2297:
2296:
2280:
2277:
2276:
2259:
2255:
2253:
2250:
2249:
2227:
2223:
2222:
2217:
2202:
2198:
2197:
2192:
2184:
2173:
2165:
2164:
2160:
2159:
2157:
2145:
2141:
2139:
2136:
2135:
2118:
2114:
2108:
2103:
2091:
2083:
2081:
2078:
2077:
2057:
2053:
2038:
2034:
2023:
2020:
2019:
2003:
2000:
1999:
1974:
1969:
1968:
1964:
1958:
1953:
1941:
1938:
1937:
1930:weak derivative
1909:
1900:
1895:
1894:
1886:
1883:
1882:
1879:
1828:
1820:Jacobian matrix
1773:
1690:smooth manifold
1678:
1658:complex numbers
1583:
1573:
1563:
1557:
1549:
1535:
1528:Jacobian matrix
1505:linear operator
1451:
1443:
1440:
1439:
1371:
1369:
1357:
1351:
1348:
1347:
1337:counterexamples
1317:
1314:
1313:
1281:
1277:
1270:
1263:
1259:
1213:
1211:
1193:
1187:
1184:
1183:
1155:
1152:
1151:
1126:
1123:
1122:
1102:
1099:
1098:
1082:
1079:
1078:
1050:
1047:
1046:
1024:
1021:
1020:
1010:
970:
941:
940:
926:Integration Bee
901:
898:
891:
890:
866:
863:
856:
855:
828:
825:
818:
817:
799:Volume integral
734:
729:
722:
721:
638:
633:
626:
625:
595:
514:
509:
502:
501:
493:Risch algorithm
463:Euler's formula
349:
344:
337:
336:
318:General Leibniz
201:generalizations
183:
178:
171:
157:Rolle's theorem
152:
127:
63:
57:
52:
46:
43:
42:
24:
17:
12:
11:
5:
5762:
5752:
5751:
5734:
5733:
5731:
5730:
5725:
5720:
5718:Choquet theory
5715:
5710:
5699:
5697:
5693:
5692:
5690:
5689:
5684:
5679:
5673:
5671:
5665:
5664:
5662:
5661:
5656:
5650:
5648:
5644:
5643:
5641:
5640:
5635:
5630:
5625:
5620:
5619:
5618:
5608:
5602:
5600:
5596:
5595:
5593:
5592:
5587:
5582:
5577:
5572:
5567:
5561:
5559:
5553:
5552:
5550:
5549:
5544:
5528:
5527:
5526:
5521:
5516:
5502:
5501:
5500:
5495:
5485:
5479:
5477:
5473:
5472:
5470:
5469:
5464:
5459:
5454:
5449:
5448:
5447:
5437:
5432:
5431:
5430:
5420:
5415:
5409:
5407:
5401:
5400:
5398:
5397:
5392:
5387:
5382:
5377:
5372:
5367:
5366:
5365:
5354:
5352:
5351:Basic concepts
5348:
5347:
5336:
5335:
5328:
5321:
5313:
5305:
5304:
5297:
5279:
5267:
5228:Hahn, Wolfgang
5219:
5208:David Hestenes
5199:
5198:
5196:
5193:
5192:
5191:
5186:
5177:
5172:
5166:
5160:
5154:
5145:
5139:
5133:
5127:
5121:
5115:
5109:
5103:
5095:
5092:
5046:
5035:characteristic
4983:measure theory
4969:locally convex
4892:
4889:
4888:
4887:
4880:
4869:
4858:
4850:
4846:
4842:
4836:
4832:
4822:
4818:
4814:
4809:
4801:
4797:
4793:
4787:
4783:
4777:
4769:
4765:
4761:
4755:
4751:
4745:
4737:
4733:
4729:
4723:
4719:
4713:
4710:
4676:
4664:
4656:
4652:
4648:
4642:
4638:
4632:
4624:
4620:
4616:
4610:
4606:
4600:
4592:
4588:
4584:
4578:
4574:
4568:
4565:
4548:given by the
4519:function space
4496:
4493:
4490:
4487:
4484:
4481:
4478:
4473:
4470:
4466:
4462:
4459:
4456:
4453:
4450:
4410:
4407:
4401:
4398:
4394:
4389:
4386:
4378:
4374:
4370:
4364:
4360:
4354:
4351:
4331:
4328:
4325:
4322:
4319:
4316:
4313:
4310:
4307:
4287:
4284:
4281:
4278:
4275:
4272:
4269:
4266:
4262:
4259:
4255:
4252:
4248:
4245:
4216:
4213:
4176:
4173:
4145:
4142:
4138:
4135:
4124:
4123:
4112:
4107:
4103:
4099:
4096:
4093:
4088:
4085:
4082:
4078:
4072:
4069:
4066:
4062:
4058:
4055:
4052:
4049:
4046:
4043:
4038:
4035:
4032:
4028:
4022:
4018:
4014:
4011:
4007:
4003:
3997:
3993:
3989:
3986:
3981:
3977:
3973:
3970:
3967:
3962:
3959:
3956:
3952:
3946:
3943:
3940:
3936:
3932:
3927:
3923:
3917:
3913:
3908:
3894:is defined by
3856:
3853:
3836:
3833:
3832:
3831:
3820:
3814:
3811:
3808:
3805:
3802:
3799:
3794:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3770:
3767:
3759:
3756:
3753:
3749:
3745:
3739:
3736:
3733:
3730:
3725:
3722:
3719:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3690:
3687:
3684:
3680:
3676:
3670:
3667:
3664:
3659:
3656:
3653:
3650:
3647:
3644:
3641:
3638:
3635:
3627:
3624:
3621:
3617:
3596:
3593:
3590:
3587:
3576:
3564:
3558:
3555:
3552:
3549:
3546:
3543:
3538:
3535:
3532:
3529:
3526:
3523:
3520:
3517:
3514:
3511:
3508:
3505:
3478:
3473:
3469:
3466:
3463:
3460:
3457:
3454:
3451:
3448:
3445:
3442:
3439:
3433:
3427:
3424:
3421:
3418:
3415:
3412:
3407:
3404:
3401:
3398:
3395:
3392:
3389:
3386:
3383:
3380:
3368:, we may have
3357:
3354:
3351:
3348:
3345:
3342:
3339:
3336:
3321:
3310:
3307:
3304:
3301:
3298:
3295:
3292:
3289:
3286:
3283:
3280:
3277:
3274:
3271:
3268:
3265:
3262:
3243:
3236:quantum groups
3176:
3170:
3167:
3164:
3161:
3158:
3155:
3150:
3147:
3144:
3141:
3138:
3135:
3132:
3129:
3126:
3123:
3117:
3114:
3111:
3108:
3105:
3100:
3096:
3078:
3075:
3061:
3057:
3054:
3051:
3048:
3028:
3025:
3022:
3019:
3016:
3013:
3010:
3007:
3004:
2983:
2979:
2976:
2955:
2951:
2947:
2943:
2940:
2929:
2928:
2917:
2913:
2907:
2904:
2900:
2895:
2891:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2867:
2864:
2861:
2857:
2852:
2846:
2843:
2840:
2836:
2821:
2820:
2808:
2803:
2799:
2796:
2793:
2790:
2787:
2784:
2781:
2778:
2775:
2772:
2769:
2765:
2759:
2756:
2752:
2747:
2741:
2738:
2735:
2731:
2706:
2686:
2683:
2679:differintegral
2640:Hessian matrix
2614:
2611:
2593:
2588:
2583:
2578:
2571:
2567:
2563:
2558:
2554:
2551:
2527:
2524:
2521:
2516:
2512:
2500:
2499:
2488:
2485:
2479:
2473:
2466:
2461:
2455:
2448:
2444:
2440:
2435:
2431:
2428:
2425:
2422:
2419:
2416:
2410:
2404:
2400:
2393:
2386:
2381:
2375:
2351:
2338:such that for
2326:
2322:
2317:
2312:
2307:
2304:
2284:
2258:
2230:
2226:
2220:
2216:
2212:
2205:
2201:
2195:
2191:
2187:
2182:
2176:
2172:
2168:
2163:
2156:
2153:
2148:
2144:
2121:
2117:
2111:
2106:
2102:
2098:
2094:
2090:
2086:
2065:
2060:
2056:
2052:
2049:
2046:
2041:
2037:
2033:
2030:
2027:
2007:
1982:
1977:
1972:
1967:
1961:
1956:
1952:
1948:
1945:
1912:
1908:
1903:
1898:
1893:
1890:
1878:
1875:
1844:vector bundles
1827:
1824:
1793:tangent vector
1772:
1769:
1742:Lie derivative
1677:
1674:
1579:
1567:
1561:
1553:
1547:
1531:
1473:
1470:
1467:
1464:
1461:
1457:
1454:
1450:
1447:
1419:
1416:
1413:
1408:
1404:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1366:
1363:
1360:
1356:
1321:
1295:
1292:
1284:
1280:
1276:
1273:
1266:
1262:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1208:
1205:
1202:
1199:
1196:
1192:
1171:
1168:
1165:
1162:
1159:
1136:
1133:
1130:
1106:
1086:
1066:
1063:
1060:
1057:
1054:
1034:
1031:
1028:
1009:
1006:
972:
971:
969:
968:
961:
954:
946:
943:
942:
939:
938:
933:
928:
923:
921:List of topics
918:
913:
908:
902:
897:
896:
893:
892:
889:
888:
883:
878:
873:
867:
862:
861:
858:
857:
852:
851:
850:
849:
844:
839:
829:
824:
823:
820:
819:
814:
813:
812:
811:
806:
801:
796:
791:
786:
781:
773:
772:
768:
767:
766:
765:
760:
755:
750:
742:
741:
735:
728:
727:
724:
723:
718:
717:
716:
715:
710:
705:
700:
695:
690:
682:
681:
677:
676:
675:
674:
669:
664:
659:
654:
649:
639:
632:
631:
628:
627:
622:
621:
620:
619:
614:
609:
604:
599:
593:
588:
583:
578:
573:
565:
564:
558:
557:
556:
555:
550:
545:
540:
535:
530:
515:
508:
507:
504:
503:
498:
497:
496:
495:
490:
485:
480:
478:Changing order
475:
465:
460:
442:
437:
432:
424:
423:
422:Integration by
419:
418:
417:
416:
411:
406:
401:
396:
386:
384:Antiderivative
378:
377:
373:
372:
371:
370:
365:
360:
350:
343:
342:
339:
338:
333:
332:
331:
330:
325:
320:
315:
310:
305:
300:
295:
290:
285:
277:
276:
270:
269:
268:
267:
262:
257:
252:
247:
242:
234:
233:
229:
228:
227:
226:
225:
224:
219:
214:
204:
191:
190:
184:
177:
176:
173:
172:
170:
169:
164:
159:
153:
151:
150:
145:
139:
138:
137:
129:
128:
116:
113:
110:
107:
104:
101:
98:
95:
92:
89:
86:
83:
79:
76:
73:
69:
66:
60:
55:
51:
41:
38:
37:
31:
30:
15:
9:
6:
4:
3:
2:
5761:
5750:
5747:
5746:
5744:
5729:
5726:
5724:
5721:
5719:
5716:
5714:
5711:
5708:
5704:
5701:
5700:
5698:
5694:
5688:
5685:
5683:
5680:
5678:
5675:
5674:
5672:
5670:
5666:
5660:
5657:
5655:
5652:
5651:
5649:
5645:
5639:
5636:
5634:
5631:
5629:
5626:
5624:
5621:
5617:
5614:
5613:
5612:
5609:
5607:
5604:
5603:
5601:
5597:
5591:
5588:
5586:
5583:
5581:
5578:
5576:
5573:
5571:
5568:
5566:
5563:
5562:
5560:
5558:
5554:
5548:
5545:
5543:
5540:
5536:
5532:
5529:
5525:
5522:
5520:
5517:
5515:
5512:
5511:
5510:
5509:set functions
5506:
5503:
5499:
5496:
5494:
5491:
5490:
5489:
5486:
5484:
5483:Besov measure
5481:
5480:
5478:
5476:Measurability
5474:
5468:
5465:
5463:
5460:
5458:
5455:
5453:
5450:
5446:
5443:
5442:
5441:
5438:
5436:
5433:
5429:
5426:
5425:
5424:
5421:
5419:
5416:
5414:
5411:
5410:
5408:
5406:
5402:
5396:
5393:
5391:
5388:
5386:
5383:
5381:
5378:
5376:
5375:Convex series
5373:
5371:
5370:Bochner space
5368:
5364:
5361:
5360:
5359:
5356:
5355:
5353:
5349:
5345:
5341:
5334:
5329:
5327:
5322:
5320:
5315:
5314:
5311:
5300:
5294:
5290:
5283:
5276:
5271:
5263:
5259:
5255:
5251:
5247:
5243:
5240:(1–2): 4–34.
5239:
5235:
5234:
5229:
5223:
5217:
5216:90-277-2561-6
5213:
5209:
5204:
5200:
5190:
5187:
5181:
5178:
5176:
5173:
5170:
5167:
5164:
5161:
5158:
5155:
5149:
5146:
5143:
5140:
5137:
5134:
5131:
5128:
5125:
5122:
5119:
5116:
5113:
5110:
5107:
5104:
5101:
5098:
5097:
5091:
5089:
5083:
5081:
5077:
5073:
5072:Banach spaces
5069:
5065:
5060:
5058:
5054:
5049:
5044:
5040:
5036:
5032:
5028:
5023:
5021:
5016:
5014:
5010:
5006:
5002:
5000:
4994:
4992:
4988:
4984:
4979:
4977:
4973:
4970:
4966:
4961:
4959:
4955:
4950:
4948:
4944:
4940:
4936:
4932:
4927:
4925:
4921:
4917:
4916:subderivative
4912:
4910:
4906:
4902:
4898:
4885:
4881:
4878:
4874:
4870:
4856:
4848:
4844:
4834:
4820:
4816:
4812:
4807:
4799:
4795:
4785:
4775:
4767:
4763:
4753:
4743:
4735:
4731:
4721:
4711:
4708:
4700:
4696:
4693:
4689:
4685:
4684:metric tensor
4681:
4680:d'Alembertian
4677:
4662:
4654:
4650:
4640:
4630:
4622:
4618:
4608:
4598:
4590:
4586:
4576:
4566:
4555:
4551:
4543:
4539:
4535:
4534:
4533:
4530:
4528:
4524:
4520:
4516:
4512:
4507:
4494:
4488:
4485:
4482:
4479:
4471:
4468:
4464:
4460:
4454:
4448:
4440:
4436:
4432:
4428:
4424:
4408:
4405:
4399:
4396:
4392:
4387:
4384:
4376:
4372:
4368:
4362:
4358:
4352:
4349:
4329:
4326:
4323:
4320:
4317:
4311:
4305:
4285:
4282:
4279:
4276:
4273:
4270:
4267:
4264:
4260:
4257:
4253:
4250:
4246:
4243:
4234:
4230:
4226:
4222:
4212:
4210:
4206:
4201:
4197:
4194:
4190:
4186:
4182:
4172:
4169:
4167:
4163:
4160:
4143:
4140:
4133:
4110:
4105:
4101:
4097:
4094:
4091:
4086:
4083:
4080:
4076:
4070:
4067:
4064:
4060:
4053:
4050:
4047:
4041:
4036:
4033:
4030:
4026:
4020:
4016:
4012:
4009:
4005:
4001:
3995:
3991:
3987:
3984:
3979:
3975:
3971:
3968:
3965:
3960:
3957:
3954:
3950:
3944:
3941:
3938:
3934:
3930:
3925:
3921:
3915:
3911:
3906:
3897:
3896:
3895:
3893:
3889:
3885:
3880:
3878:
3874:
3870:
3866:
3862:
3852:
3850:
3846:
3842:
3818:
3812:
3806:
3803:
3800:
3789:
3783:
3780:
3774:
3771:
3765:
3757:
3751:
3743:
3737:
3734:
3731:
3728:
3720:
3714:
3711:
3705:
3702:
3696:
3688:
3682:
3674:
3668:
3665:
3662:
3654:
3648:
3645:
3639:
3633:
3625:
3619:
3594:
3591:
3588:
3585:
3577:
3562:
3556:
3553:
3550:
3547:
3544:
3541:
3533:
3527:
3524:
3518:
3515:
3512:
3509:
3503:
3492:
3476:
3471:
3464:
3458:
3455:
3449:
3446:
3443:
3437:
3431:
3425:
3419:
3416:
3413:
3402:
3396:
3393:
3387:
3384:
3378:
3355:
3349:
3346:
3343:
3337:
3334:
3326:
3322:
3305:
3299:
3296:
3290:
3287:
3284:
3278:
3275:
3269:
3263:
3252:
3248:
3244:
3241:
3237:
3233:
3232:combinatorics
3229:
3225:
3221:
3217:
3213:
3212:q-deformation
3209:
3203:
3198:
3194:
3190:
3174:
3168:
3162:
3159:
3156:
3145:
3139:
3136:
3130:
3127:
3121:
3115:
3109:
3103:
3098:
3094:
3085:
3081:
3080:
3074:
3055:
3052:
3049:
3046:
3026:
3023:
3020:
3017:
3014:
3008:
3002:
2977:
2974:
2941:
2938:
2915:
2911:
2905:
2902:
2898:
2893:
2886:
2880:
2877:
2871:
2868:
2865:
2859:
2855:
2850:
2844:
2838:
2826:
2825:
2824:
2806:
2801:
2794:
2788:
2785:
2779:
2776:
2773:
2767:
2763:
2757:
2754:
2750:
2745:
2739:
2733:
2721:
2720:
2719:
2696:
2692:
2682:
2680:
2676:
2672:
2668:
2663:
2661:
2657:
2653:
2649:
2648:local extrema
2645:
2641:
2637:
2633:
2629:
2625:
2621:
2610:
2608:
2607:distributions
2591:
2586:
2576:
2565:
2556:
2552:
2549:
2541:
2525:
2522:
2519:
2514:
2510:
2486:
2483:
2477:
2471:
2464:
2453:
2442:
2429:
2426:
2420:
2417:
2414:
2408:
2402:
2398:
2391:
2384:
2373:
2365:
2364:
2363:
2349:
2341:
2315:
2305:
2302:
2282:
2256:
2228:
2224:
2218:
2214:
2210:
2203:
2199:
2193:
2189:
2180:
2170:
2154:
2151:
2146:
2142:
2119:
2115:
2109:
2104:
2100:
2096:
2088:
2058:
2054:
2050:
2047:
2044:
2039:
2035:
2028:
2025:
2005:
1997:
1996:multi-indices
1980:
1975:
1965:
1954:
1950:
1946:
1943:
1935:
1931:
1927:
1901:
1891:
1888:
1874:
1872:
1867:
1865:
1861:
1857:
1853:
1849:
1845:
1841:
1837:
1833:
1823:
1821:
1817:
1812:
1810:
1806:
1802:
1798:
1794:
1790:
1786:
1782:
1778:
1768:
1766:
1762:
1757:
1755:
1751:
1747:
1743:
1738:
1736:
1732:
1728:
1724:
1720:
1715:
1711:
1707:
1703:
1699:
1695:
1691:
1687:
1683:
1673:
1670:
1666:
1661:
1659:
1655:
1651:
1646:
1644:
1640:
1636:
1632:
1627:
1625:
1620:
1618:
1614:
1610:
1606:
1602:
1601:scalar fields
1598:
1594:
1589:
1587:
1582:
1577:
1571:
1565:
1556:
1551:
1543:
1539:
1534:
1529:
1526:known as the
1525:
1522:
1518:
1514:
1510:
1506:
1502:
1498:
1494:
1490:
1485:
1471:
1468:
1462:
1455:
1452:
1445:
1437:
1433:
1417:
1414:
1411:
1406:
1399:
1393:
1390:
1384:
1381:
1378:
1372:
1364:
1358:
1345:
1340:
1338:
1335:
1319:
1311:
1310:neighbourhood
1306:
1293:
1290:
1282:
1274:
1264:
1256:
1253:
1250:
1244:
1238:
1235:
1229:
1226:
1223:
1217:
1206:
1197:
1169:
1163:
1160:
1157:
1150:
1134:
1131:
1128:
1120:
1104:
1084:
1064:
1058:
1055:
1052:
1032:
1029:
1026:
1019:
1015:
1005:
1003:
999:
995:
994:combinatorics
991:
987:
983:
979:
967:
962:
960:
955:
953:
948:
947:
945:
944:
937:
934:
932:
929:
927:
924:
922:
919:
917:
914:
912:
909:
907:
904:
903:
895:
894:
887:
884:
882:
879:
877:
874:
872:
869:
868:
860:
859:
848:
845:
843:
840:
838:
835:
834:
833:
832:
822:
821:
810:
807:
805:
802:
800:
797:
795:
792:
790:
789:Line integral
787:
785:
782:
780:
777:
776:
775:
774:
770:
769:
764:
761:
759:
756:
754:
751:
749:
746:
745:
744:
743:
739:
738:
732:
731:Multivariable
726:
725:
714:
711:
709:
706:
704:
701:
699:
696:
694:
691:
689:
686:
685:
684:
683:
679:
678:
673:
670:
668:
665:
663:
660:
658:
655:
653:
650:
648:
645:
644:
643:
642:
636:
630:
629:
618:
615:
613:
610:
608:
605:
603:
600:
598:
594:
592:
589:
587:
584:
582:
579:
577:
574:
572:
569:
568:
567:
566:
563:
560:
559:
554:
551:
549:
546:
544:
541:
539:
536:
534:
531:
528:
524:
521:
520:
519:
518:
512:
506:
505:
494:
491:
489:
486:
484:
481:
479:
476:
473:
469:
466:
464:
461:
458:
454:
450:
449:trigonometric
446:
443:
441:
438:
436:
433:
431:
428:
427:
426:
425:
421:
420:
415:
412:
410:
407:
405:
402:
400:
397:
394:
390:
387:
385:
382:
381:
380:
379:
375:
374:
369:
366:
364:
361:
359:
356:
355:
354:
353:
347:
341:
340:
329:
326:
324:
321:
319:
316:
314:
311:
309:
306:
304:
301:
299:
296:
294:
291:
289:
286:
284:
281:
280:
279:
278:
275:
272:
271:
266:
263:
261:
260:Related rates
258:
256:
253:
251:
248:
246:
243:
241:
238:
237:
236:
235:
231:
230:
223:
220:
218:
217:of a function
215:
213:
212:infinitesimal
210:
209:
208:
205:
202:
198:
195:
194:
193:
192:
188:
187:
181:
175:
174:
168:
165:
163:
160:
158:
155:
154:
149:
146:
144:
141:
140:
136:
133:
132:
131:
130:
111:
105:
102:
96:
90:
87:
84:
81:
74:
67:
64:
58:
53:
49:
40:
39:
36:
33:
32:
28:
27:
22:
5696:Applications
5654:Crinkled arc
5590:Paley–Wiener
5451:
5288:
5282:
5270:
5237:
5231:
5222:
5203:
5169:q-derivative
5084:
5061:
5047:
5043:finite field
5033:of positive
5031:local fields
5026:
5024:
5017:
4998:
4995:
4980:
4962:
4951:
4928:
4913:
4894:
4698:
4541:
4531:
4508:
4434:
4426:
4422:
4232:
4228:
4218:
4202:
4198:
4196:either way.
4193:binary trees
4178:
4170:
4161:
4126:The mapping
4125:
3891:
3881:
3858:
3838:
3493:difference,
3239:
3223:
3207:
3201:
3196:
3192:
3191:nonzero, if
3188:
3084:q-derivative
2930:
2822:
2688:
2670:
2666:
2664:
2660:Morse theory
2638:matrix, the
2635:
2631:
2627:
2616:
2501:
2339:
1880:
1868:
1862:. See also
1829:
1813:
1804:
1781:vector field
1774:
1758:
1739:
1701:
1679:
1662:
1647:
1638:
1634:
1628:
1621:
1605:vector field
1596:
1592:
1590:
1580:
1575:
1569:
1559:
1554:
1545:
1537:
1532:
1520:
1516:
1512:
1508:
1500:
1496:
1492:
1486:
1435:
1431:
1341:
1334:pathological
1307:
1118:
1117:, is called
1011:
975:
445:Substitution
207:Differential
200:
180:Differential
5462:Holomorphic
5445:Directional
5405:Derivatives
5001:-derivative
4920:subgradient
4905:dimensional
4695:dot product
4439:integration
4181:type theory
3855:Derivations
3849:Lie algebra
3325:time scales
2695:quaternions
2362:, we have
2248:. Then the
1750:Lie algebra
1746:Lie bracket
1607:called the
978:mathematics
906:Precalculus
899:Miscellanea
864:Specialized
771:Definitions
538:Alternating
376:Definitions
189:Definitions
5057:logarithms
4550:divergence
3888:polynomial
3861:derivation
1714:divergence
1586:chain rule
1344:derivative
1182:such that
982:derivative
886:Variations
881:Stochastic
871:Fractional
740:Formalisms
703:Divergence
672:Identities
652:Divergence
197:Derivative
148:Continuity
5585:Regulated
5557:Integrals
5254:0025-584X
4960:instead.
4841:∂
4831:∂
4808:−
4792:∂
4782:∂
4760:∂
4750:∂
4728:∂
4718:∂
4709:◻
4692:Euclidean
4647:∂
4637:∂
4615:∂
4605:∂
4583:∂
4573:∂
4564:Δ
4486:−
4469:−
4406:−
4342:, where
4327:−
4283:−
4265:−
4168:as well.
4137:↦
4095:⋯
4084:−
4068:−
4051:−
4034:−
3969:⋯
3958:−
3942:−
3877:D-modules
3804:−
3781:−
3755:→
3735:−
3712:−
3686:→
3666:−
3646:−
3623:→
3554:−
3551:ω
3525:−
3519:ω
3472:ε
3456:−
3450:ε
3417:−
3394:−
3347:−
3335:ε
3297:−
3261:Δ
3160:−
3137:−
3056:∈
2978:⊂
2950:→
2903:−
2878:−
2842:→
2786:−
2755:−
2737:→
2656:manifolds
2566:α
2553:∈
2515:α
2478:φ
2454:∫
2443:α
2427:−
2409:φ
2403:α
2374:∫
2350:φ
2321:→
2257:α
2225:α
2211:⋯
2200:α
2186:∂
2181:φ
2171:α
2162:∂
2152:φ
2147:α
2116:α
2101:∑
2089:α
2055:α
2048:…
2036:α
2026:α
1960:∞
1947:∈
1944:φ
1924:which is
1907:→
1469:⋅
1449:↦
1391:−
1362:→
1279:‖
1272:‖
1261:‖
1251:−
1236:−
1215:‖
1204:→
1201:‖
1195:‖
1167:→
1132:∈
1062:→
876:Malliavin
763:Geometric
662:Laplacian
612:Dirichlet
523:Geometric
103:−
50:∫
5743:Category
5539:Strongly
5340:Analysis
5094:See also
5078:and the
5007:and the
4991:Jacobian
4554:gradient
4261:′
4247:″
4144:′
4006:′
1791:, and a
1789:manifold
1735:rotation
1706:gradient
1613:gradient
1456:′
1077:, where
1002:geometry
916:Glossary
826:Advanced
804:Jacobian
758:Exterior
688:Gradient
680:Theorems
647:Gradient
586:Integral
548:Binomial
533:Harmonic
393:improper
389:Integral
346:Integral
328:Reynolds
303:Quotient
232:Concepts
68:′
35:Calculus
5705: (
5647:Related
5599:Results
5575:Dunford
5565:Bochner
5531:Bochner
5505:Measure
5262:0030647
5068:locally
4552:of the
4189:algebra
4183:, many
3865:algebra
2995:, then
2658:, see
1856:torsion
1801:subsets
1752:of the
1688:over a
1680:On the
1641:(i.e.,
1004:, etc.
998:algebra
911:History
809:Hessian
698:Stokes'
693:Green's
525: (
447: (
391: (
313:Inverse
288:Product
199: (
5707:bundle
5535:Weakly
5524:Vector
5295:
5260:
5252:
5214:
4985:, the
4943:module
4899:, the
4513:. The
4205:zipper
2644:tensor
2481:
2475:
2412:
2395:
1994:, and
1850:. In
1840:curves
1834:, the
1723:scalar
1712:, and
1704:, the
1692:, the
1667:, the
1524:matrix
980:, the
753:Tensor
748:Matrix
635:Vector
553:Taylor
511:Series
143:Limits
5428:Total
5195:Notes
5070:like
4421:is a
3886:of a
3847:or a
2076:with
1787:on a
1633:from
1503:is a
576:Ratio
543:Power
457:Euler
435:Discs
430:Parts
298:Power
293:Chain
222:total
5293:ISBN
5250:ISSN
5212:ISBN
5025:The
4996:The
4963:The
4918:and
4914:The
4882:The
4871:The
4678:The
4536:The
3845:ring
3491:Hahn
3245:The
3218:and
3187:For
3082:The
3039:for
1869:The
1814:The
1779:, a
1740:The
1727:flux
1710:curl
1629:For
1622:The
1566:) =J
1012:The
657:Curl
617:Abel
581:Root
5342:in
5242:doi
4981:In
4952:In
4941:or
4929:In
4911:.
4895:In
4697:of
4686:of
4179:In
3748:lim
3679:lim
3616:lim
3249:of
3204:→ 1
2835:lim
2730:lim
2689:In
2634:by
2340:all
1846:or
1830:In
1803:of
1775:In
1729:by
1721:of
1684:of
1663:In
1648:In
1637:to
1595:to
1519:by
1507:on
1495:to
1487:In
1355:lim
1312:of
1191:lim
1121:at
976:In
283:Sum
5745::
5537:/
5533:/
5258:MR
5256:.
5248:.
5236:.
5090:.
5082:.
5055:,
5015:.
4978:.
4933:,
4701::
4525:,
4219:A
4211:.
3859:A
3851:.
3073:.
2681:.
2662:.
2523::=
2261:th
2155::=
2097::=
1822:.
1811:.
1767:.
1708:,
1619:.
1588:.
1578:)J
1568:ƒ(
1484:.
1339:.
1294:0.
1000:,
996:,
992:,
455:,
451:,
5709:)
5332:e
5325:t
5318:v
5301:.
5264:.
5244::
5238:2
5048:q
5045:F
4999:H
4857:.
4849:2
4845:t
4835:2
4821:2
4817:c
4813:1
4800:2
4796:z
4786:2
4776:+
4768:2
4764:y
4754:2
4744:+
4736:2
4732:x
4722:2
4712:=
4699:R
4663:.
4655:2
4651:z
4641:2
4631:+
4623:2
4619:y
4609:2
4599:+
4591:2
4587:x
4577:2
4567:=
4546:Δ
4542:R
4495:.
4492:)
4489:1
4483:x
4480:4
4477:(
4472:1
4465:L
4461:=
4458:)
4455:x
4452:(
4449:f
4435:x
4427:x
4409:3
4400:x
4397:d
4393:d
4388:2
4385:+
4377:2
4373:x
4369:d
4363:2
4359:d
4353:=
4350:L
4330:1
4324:x
4321:4
4318:=
4315:)
4312:f
4309:(
4306:L
4286:1
4280:x
4277:4
4274:=
4271:f
4268:3
4258:f
4254:2
4251:+
4244:f
4233:x
4231:(
4229:f
4162:R
4141:f
4134:f
4111:.
4106:1
4102:a
4098:+
4092:+
4087:2
4081:d
4077:x
4071:1
4065:d
4061:a
4057:)
4054:1
4048:d
4045:(
4042:+
4037:1
4031:d
4027:x
4021:d
4017:a
4013:d
4010:=
4002:)
3996:0
3992:a
3988:+
3985:x
3980:1
3976:a
3972:+
3966:+
3961:1
3955:d
3951:x
3945:1
3939:d
3935:a
3931:+
3926:d
3922:x
3916:d
3912:a
3907:(
3892:R
3819:.
3813:x
3810:)
3807:1
3801:q
3798:(
3793:)
3790:x
3787:(
3784:f
3778:)
3775:x
3772:q
3769:(
3766:f
3758:1
3752:q
3744:=
3738:x
3732:x
3729:q
3724:)
3721:x
3718:(
3715:f
3709:)
3706:x
3703:q
3700:(
3697:f
3689:1
3683:q
3675:=
3669:x
3663:z
3658:)
3655:x
3652:(
3649:f
3643:)
3640:z
3637:(
3634:f
3626:x
3620:z
3595:x
3592:q
3589:=
3586:z
3563:.
3557:x
3548:+
3545:x
3542:q
3537:)
3534:x
3531:(
3528:f
3522:)
3516:+
3513:x
3510:q
3507:(
3504:f
3477:.
3468:)
3465:x
3462:(
3459:f
3453:)
3447:+
3444:x
3441:(
3438:f
3432:=
3426:x
3423:)
3420:1
3414:q
3411:(
3406:)
3403:x
3400:(
3397:f
3391:)
3388:x
3385:q
3382:(
3379:f
3356:x
3353:)
3350:1
3344:q
3341:(
3338:=
3309:)
3306:x
3303:(
3300:f
3294:)
3291:1
3288:+
3285:x
3282:(
3279:f
3276:=
3273:)
3270:x
3267:(
3264:f
3240:q
3224:q
3208:q
3202:q
3197:x
3193:f
3189:x
3175:.
3169:x
3166:)
3163:1
3157:q
3154:(
3149:)
3146:x
3143:(
3140:f
3134:)
3131:x
3128:q
3125:(
3122:f
3116:=
3113:)
3110:x
3107:(
3104:f
3099:q
3095:D
3060:H
3053:b
3050:,
3047:a
3027:b
3024:q
3021:+
3018:a
3015:=
3012:)
3009:q
3006:(
3003:f
2982:H
2975:U
2954:H
2946:H
2942::
2939:f
2916:.
2912:]
2906:1
2899:h
2894:)
2890:)
2887:a
2884:(
2881:f
2875:)
2872:h
2869:+
2866:a
2863:(
2860:f
2856:(
2851:[
2845:0
2839:h
2807:]
2802:)
2798:)
2795:a
2792:(
2789:f
2783:)
2780:h
2777:+
2774:a
2771:(
2768:f
2764:(
2758:1
2751:h
2746:[
2740:0
2734:h
2705:H
2671:n
2667:n
2636:n
2632:n
2628:n
2592:)
2587:n
2582:R
2577:(
2570:|
2562:|
2557:C
2550:u
2526:v
2520:u
2511:D
2487:x
2484:d
2472:v
2465:n
2460:R
2447:|
2439:|
2434:)
2430:1
2424:(
2421:=
2418:x
2415:d
2399:D
2392:u
2385:n
2380:R
2325:R
2316:n
2311:R
2306::
2303:v
2283:u
2229:n
2219:n
2215:x
2204:1
2194:1
2190:x
2175:|
2167:|
2143:D
2120:i
2110:n
2105:1
2093:|
2085:|
2064:)
2059:n
2051:,
2045:,
2040:1
2032:(
2029:=
2006:n
1981:)
1976:n
1971:R
1966:(
1955:c
1951:C
1911:R
1902:n
1897:R
1892::
1889:u
1805:R
1702:R
1639:R
1635:R
1599:(
1597:R
1593:R
1581:x
1576:g
1574:(
1572:)
1570:x
1564:f
1562:°
1560:g
1558:(
1555:x
1550:f
1548:°
1546:g
1538:x
1533:x
1530:J
1521:n
1517:m
1509:R
1501:A
1497:R
1493:R
1472:t
1466:)
1463:x
1460:(
1453:f
1446:t
1436:A
1432:A
1418:,
1415:A
1412:=
1407:h
1403:)
1400:x
1397:(
1394:f
1388:)
1385:h
1382:+
1379:x
1376:(
1373:f
1365:0
1359:h
1320:x
1291:=
1283:V
1275:h
1265:W
1257:h
1254:A
1248:)
1245:x
1242:(
1239:f
1233:)
1230:h
1227:+
1224:x
1221:(
1218:f
1207:0
1198:h
1170:W
1164:V
1161::
1158:A
1135:U
1129:x
1105:V
1085:U
1065:W
1059:U
1056::
1053:f
1033:W
1030:,
1027:V
965:e
958:t
951:v
529:)
474:)
470:(
459:)
395:)
203:)
115:)
112:a
109:(
106:f
100:)
97:b
94:(
91:f
88:=
85:t
82:d
78:)
75:t
72:(
65:f
59:b
54:a
23:.
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