- Derivative (generalizations)
Derivative is a fundamental construction ofdifferential calculus and admits many possible generalizations within the fields ofmathematical analysis ,combinatorics ,algebra , andgeometry .**Derivatives in analysis**In real, complex, and functional analysis, derivatives are generalized to functions of several real or complex variables and functions between

topological vector spaces . An important case is the variational derivative in thecalculus of variations . Repeated application of differentiation leads to derivatives of higher order and differential operators.**Multivariable calculus**The derivative is often met for the first time as an operation on a single real function of a single real variable. One of the simplest settings for generalizations is to vector valued functions of several variables (most often the domain forms a vector space as well). This is the field of

multivariable calculus .In one-variable calculus, we say that a function is

**differentiable**at a point "x" if the limit:$lim\_\{h\; o\; 0\}frac\{f(x+h)\; -\; f(x)\}\{h\}$exists, its value is then the derivative ƒ'("x"). A function is differentiable on an interval if it is differentiable at every point within the interval.We can generalize to functions mapping

**R**^{"m"}to**R**^{"n"}as follows: ƒ is differentiable at "x" if there exists alinear operator "A"("x") (depending on "x") such that:$lim\_\; =\; 0.$Note that, in general, we concern ourselves mostly with functions being differentiable in some open neighbourhood of $x$ rather than at individual points, as not doing so tends to lead to many pathologicalcounterexamples .An "m" by "n" matrix, of the

linear operator "A"("x") is known asmatrixJacobian **J**_{"x"}(ƒ) of the mapping ƒ at point "x". Each entry of this matrix represents a, specifying the rate of change of one range coordinate with respect to a change in a domain coordinate. Of course, the Jacobianmatrix of the composition "gpartial derivative _{°}f" is a product of corresponding Jacobian matrices:**J**_{"x"}("g_{°}f") =**J**_{ƒ("x")}("g")**J**_{"x"}(ƒ).For real valued functions from

**R**^{"n"}to**R**(scalar field s), the total derivative can be interpreted as avector field called the. 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 calculategradient ofdirectional derivative sscalar functions or normal directions.Several linear combinations of partial derivatives are especially useful in the context of differential equations defined by a vector valued function

**R**^{"n"}to**R**^{"n"}. Thegives a measure of how much "source" or "sink" near a point there is. It can be used to calculatedivergence flux bydivergence theorem . The**curl**measures how much "rotation " a vector field has near a point.For

vector-valued functions from**R**to**R**^{"n"}(i.e.,parametric curve s), one can take the derivative of each component separately. The resulting derivative is another vector valued function. 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.The

takes into account changes due to time dependence and motion through space along vector field.convective derivative **Convex analysis**The

subderivative andsubgradient are generalizations of the derivative toconvex function s.**Higher-order derivatives and differential operators**One can iterate the differentiation process, that is, apply derivatives more than once, obtaining derivatives of second and higher order. A more sophisticated idea is to combine several derivatives, possibly of different orders, in one algebraic expression, a

differential operator . This is especially useful in considering ordinarylinear differential equation s with constant coefficients. For example, if "f"("x") is a twice differentiable function of one variable, the differential equation: $f"+2f\text{\'}-3f=4x-1,$

may be rewritten in the form

: $L(f)=4x-1,,$    where    $L=frac\{d^2\}\{dx^2\}+2frac\{d\}\{dx\}-3$

is a "second order linear constant coefficient differential operator" acting on functions of "x". The key idea here is that we consider a particular

linear combination 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"x" − 1, by analogy with ordinary integration, and formally write: $f(x)=L^\{-1\}(4x-1).,$

Higher derivatives can also be defined for functions of several variables, studied in in

multivariable calculus . In this case, instead of repeatedly applying the derivative, one repeatedly appliespartial derivative s with respect to different variables. For example, the second order partial derivatives of a scalar function of "n" variables can be organized into an "n" by "n" matrix, the. 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 aHessian matrix tensor ). Nevertheless, higher derivatives have important applications to analysis of local extrema of a function at its critical points. For an advanced application of this analysis to topology ofmanifold s, seeMorse theory .As in the case of functions of one variable, we can combine first and higher order partial derivatives to arrive at a notion of a

partial differential operator . Some of these operators are so important that they have their own names:*The

Laplace operator or**Laplacian**on**R**^{3}is a second-order partial differential operator "Δ" given by thedivergence of thegradient of a scalar function of three variables, or explicitly as:: $Delta=frac\{partial^2\}\{partial\; x^2\}+frac\{partial^2\}\{partial\; y^2\}+frac\{partial^2\}\{partial\; z^2\}.$Analogous operators can be defined for functions of any number of variables.

*The

d'Alembertian or**wave operator**is similar to the Laplacian, but acts on functions of four variables. Its definition uses the indefinite metric ofMinkowski space , instead of the Euclideandot product of**R**^{"3"}::: $square=frac\{partial^2\}\{partial\; x^2\}+frac\{partial^2\}\{partial\; y^2\}+frac\{partial^2\}\{partial\; z^2\}-frac\{1\}\{c^2\}frac\{partial^2\}\{partial\; t^2\}.$

**Analysis on fractals**Laplacians and differential equations can be defined on fractals.

**Fractional derivatives**In addition to "n"-th derivatives for any natural number "n", there are various ways to define derivatives of fractional or negative orders, which are studied in

. The -1 order derivative corresponds to the integral, whence the termfractional calculus .differintegral **Complex analysis**In

complex analysis , the central objects of study areholomorphic functions , which are complex-valued functions on thecomplex numbers satisfying a suitably extended definition of differentiability.The

describes how a complex function is approximated by aSchwarzian derivative fractional-linear map , in much the same way that a normal derivative describes how a function is approximated by a linear map.**Functional analysis**In

functional analysis , thedefines 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 infinitefunctional derivative dimension al vector space.The

allows the extension of the directional derivative to a generalFréchet derivative Banach space . Theextends the concept toGâteaux derivative locally convex topological vector space s. Fréchet differentiability is a strictly stronger condition than Gâteaux differentiability, even in finite dimensions. Between the two extremes is the.quasi-derivative In

measure theory , thegeneralizes theRadon-Nikodym derivative Jacobian , used for changing variables, to measures. It expresses one measure μ in terms of another measure ν (under certain conditions).In the theory of

abstract Wiener space s, the "H"-derivative defines a derivative in certain directions corresponding to the Cameron-MartinHilbert space .The derivative also admits a generalization to the space of

**distributions**on a space of functions usingintegration by parts against a suitably well-behaved subspace.On a

function space , thelinear operator which assigns to each function its derivative is an example of a. General differential operators include higher order derivatives. By means of thedifferential operator Fourier transform ,can be defined which allow for fractional calculus.pseudo-differential operator s**Difference operator, q-analogues and time scales*** The

of a function is defined by the formulaq-derivative : $D\_q\; f(x)=frac\{f(qx)-f(x)\}\{(q-1)x\}.$

If "f" is a differentiable function of "x" then in the limit as "q" → 1 we obtain the ordinary derivative, thus the "q"-derivative may be viewed as its

q-deformation . A large body of results from ordinary differential calculus, such asbinomial formula andTaylor expansion , have natural "q"-analogues that were discovered in the 19th century, but remained relatively obscure for a big part of the 20th century, outside of the theory ofspecial functions . The progress ofcombinatorics and the discovery ofquantum group s have changed the situation dramatically, and the popularity of "q"-analogues is on the rise.* The

ofdifference operator difference equations is another discrete analog of the standard derivative.:$Delta\; f(x)=f(x+1)-f(x),$* The

**q-derivative**, the**difference operator**and the**standard derivative**can all be viewed as the same thing on different time scales.**Derivatives in algebra**In algebra, generalizations of the derivative can be obtained by imposing the Leibnitz rule of differentiation in an algebraic structure, such as a ring or a

Lie algebra .**Derivations**A

**derivation**is a linear map on a ring or algebra which satisfies the Leibnitz law (the product rule). Higher derivatives and algebraic differential operators can also be defined. They are studied in a purely algebraic setting indifferential Galois theory and the theory ofD-module s, but also turn up in many other areas, where they often agree with less algebraic definitions of derivatives.For example, the

**formal derivative**of apolynomial over a commutative ring "R" is defined by:$(a\_dx^d\; +\; a\_\{d-1\}x^\{d-1\}\; +\; cdots+a\_1x+a\_0)\text{'}\; =\; da\_dx^\{d-1\}+(d-1)a\_\{d-1\}x^\{d-2\}\; +\; cdots+a\_1.$The mapping $fmapsto\; f\text{'}$ is then a derivation on thepolynomial ring "R" ["X"] . This definition can be extended torational function s as well.The notion of derivation applies to noncommutative as well as commutative rings, and even to non-associative algebraic structures, such as Lie algebras.

Also see

Pincherle derivative .**Commutative algebra**In

commutative algebra ,are universal derivations of aKähler differential scommutative ring or module. They can be used to define an analogue of exterior derivative from differential geometry that applies to arbitraryalgebraic varieties , instead of just smooth manifolds.**Number theory**In

p-adic analysis , the usual definition of derivative is not quite strong enough, and one requires strict differentiability instead.Also see

arithmetic derivative .**Set theory and logic**The ideas of zero, addition, multiplication and exponentiation known from the area of arithmetic have analogies in set theory, category theory and type theory. For example, there follows a small list of set theoretical ones:

*Theempty set ::$varnothing$

*Thecartesian product of the sets $A$ and $B$::$A\; imes\; B$

*the set of $n$-tuples of elements of the set $A$::$A^n;forall\; ninmathbb\; N$

*the set of functions from $A$ to $B$ ::$B^A,$ (sometimes written as $A\; o\; B$ or $mathrm\{Hom\}(A,B),$)Also::$A\; +\; B,$can be defined for sets as a fruitful concept. It something similar to the disjoint union of sets, but it uses labels to achieve a partition-like construct [*More precisely, it is a two-arguments case of the more general construct::$sum\_Lambda\; vec\; A\; =\; left\{leftlanglelambda,\; a\; ight\; angle\; in\; Lambda\; imes\; igcupmathcal\; A\; mid\; lambdainLambda\; land\; a\; in\; vec\; A\_lambda\; ight\}$ where $vec\; A\; :\; Lambda\; o\; mathcal\; A$*] .There are analogous constructs for types, too (see also typeful functional programming languages). Now let us see parametric types, e.g.::$F(X)\; =\; X^3,$Thus, we can write “polynomials” for types. Let us define the derivative here as we define it for polynomials over a ring::$F^prime(X)\; =\; X^2\; +\; X^2\; +\; X^2$The given expession can represent a (homogenous) triple “with a hole”.There are also more interesting constructs, than such polynomial ones. This notion of “derivative” can be extended also to them.This concept has practical applications (in functional programming); for example, see

Zipper (data structure) .**Derivatives in geometry**Main types of derivatives in geometry are Lie derivatives along a vector field, exterior differential, and covariant derivatives.

**Differential topology**In

differential topology , amay be defined as a derivation on the ring ofvector field smooth function s on amanifold , and amay be defined as a derivation at a point. This allows the abstraction of the notion of atangent vector directional derivative of a scalar function to general manifolds. For manifolds that aresubset s of**R**^{"n"}, this tangent vector will agree with the directional derivative defined above.The

**differential or pushforward**of a map between manifolds is the induced map between tangent spaces of those maps. It abstracts theJacobian matrix .On the

exterior algebra ofdifferential forms over asmooth manifold , theis the unique linear map which satisfies aexterior derivative graded version of the Leibniz law and squares to zero. It is a grade 1 derivation on the exterior algebra.The

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 aLie derivative Lie bracket (vector fields form theLie algebra of thediffeomorphism group of the manifold). It is a grade 0 derivation on the algebra.Together with the

(a degree -1 derivation on the exterior algebra defined by contraction with a vector field), the exterior derivative and the Lie derivative form ainterior product Lie superalgebra .**Differential geometry**In

differential geometry , themakes a choice for taking directional derivatives of vector fields alongcovariant derivative curve s. This extends the directional derivative of scalar functions to sections ofvector bundle s orprincipal bundle s. InRiemannian geometry , the existence of a metric chooses a unique preferredtorsion -free covariant derivative, known as theLevi-Civita connection . See alsogauge covariant derivative for a treatment oriented to physics.The

extends the exterior derivative to vector valued forms.exterior covariant derivative **Other generalizations**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

Finsler geometry , one studies spaces which looklocally likeBanach space s. Thus one might want a derivative with some of the features of afunctional derivative and thecovariant derivative .The study of

stochastic processes requires a form of calculus known as theMalliavin calculus . One notion of derivative in this setting is the "H"-derivative of a function on anabstract Wiener space .Also see

arithmetic derivative .**Notes**

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