Derivative (generalizations)
Encyclopedia
The derivative
is a fundamental construction of differential calculus
and admits many possible generalizations within the fields of mathematical analysis
, combinatorics
, algebra
, and geometry
.
in the calculus of variations
. Repeated application of differentiation leads to derivatives of higher order and differential operators.
.
In one-variable calculus, we say that a function
is differentiable at a point x if the limit
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. Since the line
is tangent to the original function at the point 
, the derivative can be seen as a way to find the best linear approximation of a function. If one ignores the constant term, setting 
, L(z) becomes an actual linear operator on R considered as a vector space over itself.
This motivates the following generalization to functions mapping Rm to Rn: ƒ is differentiable at x if there exists a linear operator A(x) (depending on x) such that
Although this definition is perhaps not as explicit as the above, if such an operator exists, then it is unique, and in the one-dimensional case coincides with the original definition. (In this case the derivative is represented by a 1-by-1 matrix consisting of the sole entry f(x).) Note that, in general, we concern ourselves mostly with functions being differentiable in some open neighbourhood
of
rather than at individual points, as not doing so tends to lead to many pathological
counterexamples.
An m by n matrix
, of the linear operator A(x) is known as Jacobian matrix Jx(ƒ) of the mapping ƒ at point x. Each entry of this matrix represents a partial derivative
, 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 g°f is a product of corresponding Jacobian matrices:
Jx(g°f) =Jƒ(x)(g)Jx(ƒ). This is a higher-dimensional statement of the chain rule
.
For real valued functions from Rn to R (scalar field
s), the total derivative can be interpreted as a vector field
called the gradient
. 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 directional derivative
s of scalar
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 Rn to Rn. The divergence
gives a measure of how much "source" or "sink" near a point there is. It can be used to calculate flux
by divergence theorem
. The curl measures how much "rotation
" a vector field has near a point.
For vector-valued functions from R to Rn (i.e., parametric curves), 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 convective derivative
takes into account changes due to time dependence and motion through space along vector field.
and subgradient are generalizations of the derivative to convex function
s.
. This is especially useful in considering ordinary linear differential equation
s with constant coefficients. For example, if f(x) is a twice differentiable function of one variable, the differential equation
may be rewritten in the form
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 4x − 1, by analogy with ordinary integration
, and formally write
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 applies partial 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 Hessian matrix
. 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 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 of manifold
s, see Morse 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:
Analogous operators can be defined for functions of any number of variables.
.
. The -1 order derivative corresponds to the integral, whence the term differintegral.
, the central objects of study are holomorphic function
s, which are complex-valued functions on the complex numbers satisfying a suitably extended definition of differentiability
.
The Schwarzian derivative
describes how a complex function is approximated by a fractional-linear map, in much the same way that a normal derivative describes how a function is approximated by a linear map.
, the functional derivative
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 dimension
al vector space.
The Fréchet derivative
allows the extension of the directional derivative to a general Banach space
. The Gâteaux derivative
extends the concept to 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, the Radon–Nikodym derivative generalizes the 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-Martin Hilbert space
.
The derivative also admits a generalization to the space of distributions
on a space of functions using integration by parts
against a suitably well-behaved subspace.
On a function space
, the linear operator which assigns to each function its derivative is an example of a differential operator
. General differential operators include higher order derivatives. By means of the Fourier transform
, pseudo-differential operator
s can be defined which allow for fractional calculus.
in the form of formal Laurent series with coefficients in some finite field
Fq (it is known that any local field of positive characteristic is isomorphic to a Laurent series field).
Along with suitably defined analogs to the exponential function
, logarithms and others the derivative can be used to develop notions of smoothness, analycity, integration, Taylor series as well as a theory of differential equations.
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 as binomial formula and Taylor 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 of special functions
. The progress of combinatorics
and the discovery of quantum group
s have changed the situation dramatically, and the popularity of q-analogues is on the rise.
in an algebraic structure, such as a ring
or a Lie algebra
.
is a linear map on a ring or algebra
which satisfies the Leibniz law (the product rule). Higher derivatives and algebraic differential operators
can also be defined. They are studied in a purely algebraic setting in differential Galois theory
and the theory of D-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 a polynomial
over a commutative ring R is defined by
The mapping
is then a derivation on the polynomial ring
R[X]. This definition can be extended to rational 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.
, Kähler differential
s are universal derivations of a commutative ring
or module. They can be used to define an analogue of exterior derivative
from differential geometry that applies to arbitrary algebraic varieties, instead of just smooth manifolds.
, the usual definition of derivative is not quite strong enough, and one requires strict differentiability
instead.
Also see arithmetic derivative
.
s in mathematics and computer science
can be described as the algebra
generated by a transformation that maps structures based on the type back into the type. For example, the type T of binary tree
s containing values of type A can be represented as the algebra generated by the transformation 1+A×T2→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 either way.
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 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.
This concept of a derivative of a type has practical applications, such as the zipper
technique used in functional programming languages.
, a vector field
may be defined as a derivation on the ring of smooth function
s on a manifold
, and a tangent vector
may be defined as a derivation at a point. This allows the abstraction of the notion of a directional derivative
of a scalar function to general manifolds. For manifolds that are subset
s of Rn, 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 the Jacobian matrix.
On the exterior algebra
of differential forms over a smooth manifold, the exterior derivative
is the unique linear map which satisfies a graded version of the Leibniz law and squares to zero. It is a grade 1 derivation on the exterior algebra.
The Lie derivative
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 Lie bracket
(vector fields form the Lie algebra
of the diffeomorphism group of the manifold). It is a grade 0 derivation on the algebra.
Together with the interior product (a degree -1 derivation on the exterior algebra defined by contraction with a vector field), the exterior derivative and the Lie derivative form a Lie superalgebra
.
makes a choice for taking directional derivatives of vector fields along curve
s. This extends the directional derivative of scalar functions to sections of vector bundle
s or principal bundle
s. In Riemannian geometry
, the existence of a metric chooses a unique preferred torsion
-free covariant derivative, known as the Levi-Civita connection
. See also gauge covariant derivative
for a treatment oriented to physics.
The exterior covariant derivative
extends the exterior derivative to vector valued forms.
s. Thus one might want a derivative with some of the features of a functional derivative
and the covariant derivative
.
The study of stochastic processes requires a form of calculus known as the Malliavin calculus
. One notion of derivative in this setting is the H-derivative of a function on an abstract Wiener space
.
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
is a fundamental construction of differential calculus
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....
and admits many possible generalizations within the fields of mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
, algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
, and geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
.
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 derivativeFunctional derivative
In mathematics and theoretical physics, the functional derivative is a generalization of the gradient. While the latter differentiates with respect to a vector with discrete components, the former differentiates with respect to a continuous function. Both of these can be viewed as extensions of...
in the calculus of variations
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
. 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 calculusMultivariable calculus
Multivariable calculus is the extension of calculus in one variable to calculus in more than one variable: the differentiated and integrated functions involve multiple variables, rather than just one....
.
In one-variable calculus, we say that a function
is differentiable at a point x if the limitexists. Its value is then the derivative ƒ'(x). A function is differentiable on an interval
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
if it is differentiable at every point within the interval. Since the line
is tangent to the original function at the point , the derivative can be seen as a way to find the best linear approximation of a function. If one ignores the constant term, setting , L(z) becomes an actual linear operator on R considered as a vector space over itself.This motivates the following generalization to functions mapping Rm to Rn: ƒ is differentiable at x if there exists a linear operator A(x) (depending on x) such that
Although this definition is perhaps not as explicit as the above, if such an operator exists, then it is unique, and in the one-dimensional case coincides with the original definition. (In this case the derivative is represented by a 1-by-1 matrix consisting of the sole entry f(x).) Note that, in general, we concern ourselves mostly with functions being differentiable in some open neighbourhood
Neighbourhood (mathematics)
In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.This concept is closely related to the...
of
rather than at individual points, as not doing so tends to lead to many pathologicalPathological (mathematics)
In mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive; the opposite is well-behaved....
counterexamples.
An m by n matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
, of the linear operator A(x) is known as Jacobian matrix Jx(ƒ) of the mapping ƒ at point x. Each entry of this matrix represents a partial derivative
Partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...
, 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 g°f is a product of corresponding Jacobian matrices:
Jx(g°f) =Jƒ(x)(g)Jx(ƒ). This is a higher-dimensional statement of the chain rule
Chain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...
.
For real valued functions from Rn to R (scalar field
Scalar field
In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...
s), the total derivative can be interpreted as a vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
called the gradient
Gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
. 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 directional derivative
Directional derivative
In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P in the direction of V...
s of scalar
Scalar (mathematics)
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
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 Rn to Rn. The divergence
Divergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
gives a measure of how much "source" or "sink" near a point there is. It can be used to calculate flux
Flux
In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...
by divergence theorem
Divergence theorem
In vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem , or Gauss–Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, the divergence theorem...
. The curl measures how much "rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
" a vector field has near a point.
For vector-valued functions from R to Rn (i.e., parametric curves), 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 convective derivative
Convective derivative
The material derivative is a derivative taken along a path moving with velocity v, and is often used in fluid mechanics and classical mechanics...
takes into account changes due to time dependence and motion through space along vector field.
Convex analysis
The subderivativeSubderivative
In mathematics, the concepts of subderivative, subgradient, and subdifferential arise in convex analysis, that is, in the study of convex functions, often in connection to convex optimization....
and subgradient are generalizations of the derivative to convex function
Convex function
In mathematics, a real-valued function f defined on an interval is called convex if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph is a convex set...
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 operatorDifferential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
. This is especially useful in considering ordinary linear differential equation
Linear differential equation
Linear differential equations are of the formwhere the differential operator L is a linear operator, y is the unknown function , and the right hand side ƒ is a given function of the same nature as y...
s with constant coefficients. For example, if f(x) is a twice differentiable function of one variable, the differential equation
may be rewritten in the form
-
where 
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
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
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 4x − 1, by analogy with ordinary integration
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
, and formally write
Higher derivatives can also be defined for functions of several variables, studied in in multivariable calculus
Multivariable calculus
Multivariable calculus is the extension of calculus in one variable to calculus in more than one variable: the differentiated and integrated functions involve multiple variables, rather than just one....
. In this case, instead of repeatedly applying the derivative, one repeatedly applies partial derivative
Partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...
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 Hessian matrix
Hessian matrix
In mathematics, the Hessian matrix is the square matrix of second-order partial derivatives of a function; that is, it describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named...
. 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 tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
). Nevertheless, higher derivatives have important applications to analysis of local extrema
Maxima and minima
In mathematics, the maximum and minimum of a function, known collectively as extrema , are the largest and smallest value that the function takes at a point either within a given neighborhood or on the function domain in its entirety .More generally, the...
of a function at its critical points
Critical point (mathematics)
In calculus, a critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is 0. The value of the function at a critical point is a critical value of the function...
. For an advanced application of this analysis to topology of manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s, see Morse theory
Morse theory
In differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a differentiable function on a manifold will, in a typical case, reflect...
.
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 operatorLaplace operatorIn mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or Δ...
or Laplacian on R3 is a second-order partial differential operator Δ given by the divergenceDivergenceIn vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
of the gradientGradientIn vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
of a scalar function of three variables, or explicitly as
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 tensorMetric tensorIn the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
of Minkowski spaceMinkowski spaceIn physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
, instead of the EuclideanEuclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
dot productDot productIn mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
of R3:
Analysis on fractals
Laplacians and differential equations can be defined on fractalsAnalysis on fractals
Analysis on fractals or calculus on fractals is a generalization of calculus on smooth manifolds to calculus on fractals.The theory describes dynamical phenomena which occur on objects modelled by 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 fractional calculusFractional calculus
Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers or complex number powers of the differentiation operator.and the integration operator J...
. The -1 order derivative corresponds to the integral, whence the term differintegral.
Complex analysis
In complex analysisComplex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
, the central objects of study are holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
s, which are complex-valued functions on the complex numbers satisfying a suitably extended definition of differentiability
Fréchet derivative
In mathematics, the Fréchet derivative is a derivative defined on Banach spaces. Named after Maurice Fréchet, it is commonly used to formalize the concept of the functional derivative used widely in the calculus of variations. Intuitively, it generalizes the idea of linear approximation from...
.
The Schwarzian derivative
Schwarzian derivative
In mathematics, the Schwarzian derivative, named after the German mathematician Hermann Schwarz, is a certain operator that is invariant under all linear fractional transformations. Thus, it occurs in the theory of the complex projective line, and in particular, in the theory of modular forms and...
describes how a complex function is approximated by a 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 analysisFunctional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, the functional derivative
Functional derivative
In mathematics and theoretical physics, the functional derivative is a generalization of the gradient. While the latter differentiates with respect to a vector with discrete components, the former differentiates with respect to a continuous function. Both of these can be viewed as extensions of...
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 dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
al vector space.
The Fréchet derivative
Fréchet derivative
In mathematics, the Fréchet derivative is a derivative defined on Banach spaces. Named after Maurice Fréchet, it is commonly used to formalize the concept of the functional derivative used widely in the calculus of variations. Intuitively, it generalizes the idea of linear approximation from...
allows the extension of the directional derivative to a general Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
. The Gâteaux derivative
Gâteaux derivative
In mathematics, the Gâteaux differential or Gâteaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gâteaux, a French mathematician who died young in World War I, it is defined for functions between locally convex topological vector...
extends the concept to locally convex topological vector space
Topological vector space
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
s. Fréchet differentiability is a strictly stronger condition than Gâteaux differentiability, even in finite dimensions. Between the two extremes is the quasi-derivative
Quasi-derivative
In mathematics, the quasi-derivative is one of several generalizations of the derivative of a function between two Banach spaces. The quasi-derivative is a slightly stronger version of the Gâteaux derivative, though weaker than the Fréchet derivative....
.
In measure theory, the Radon–Nikodym derivative generalizes the 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
Abstract Wiener space
An abstract Wiener space is a mathematical object in measure theory, used to construct a "decent" measure on an infinite-dimensional vector space. It is named after the American mathematician Norbert Wiener...
s, the H-derivative defines a derivative in certain directions corresponding to the Cameron-Martin Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
.
The derivative also admits a generalization to the space of distributions
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
on a space of functions using integration by parts
Integration by parts
In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals...
against a suitably well-behaved subspace.
On a function space
Function space
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...
, the linear operator which assigns to each function its derivative is an example of a differential operator
Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
. General differential operators include higher order derivatives. By means of the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
, pseudo-differential operator
Pseudo-differential operator
In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory....
s can be defined which allow for fractional calculus.
Analogues of derivatives in fields of positive characteristic
The Carlitz derivative is an operation similar to usual differentiation have been devised with the usual context of real or complex numbers changed to local fields of positive characteristicCharacteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
in the form of formal Laurent series with coefficients in some finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
Fq (it is known that any local field of positive characteristic is isomorphic to a Laurent series field).
Along with suitably defined analogs to the exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
, logarithms and others the derivative can be used to develop notions of smoothness, analycity, integration, Taylor series as well as a theory of differential equations.
Difference operator, q-analogues and time scales
- The q-derivativeQ-derivativeIn mathematics, in the area of combinatorics, the q-derivative, or Jackson derivative, is a q-analog of the ordinary derivative, introduced by Frank Hilton Jackson. It is the inverse of Jackson's q-integration-Definition:...
of a function is defined by the formula
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 as binomial formula and Taylor 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 of special functions
Special functions
Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications....
. The progress of combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
and the discovery of quantum group
Quantum group
In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra...
s have changed the situation dramatically, and the popularity of q-analogues is on the rise.
- The difference operator of difference equations is another discrete analog of the standard derivative.
- The q-derivative, the difference operator and the standard derivative can all be viewed as the same thing on different time scalesTime scale calculusIn mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying hybrid discrete–continuous dynamical systems...
.
Derivatives in algebra
In algebra, generalizations of the derivative can be obtained by imposing the Leibniz rule of differentiationProduct rule
In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated thus:'=f'\cdot g+f\cdot g' \,\! or in the Leibniz notation thus:...
in an algebraic structure, such as a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
or a Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
.
Derivations
A derivationDerivation (abstract algebra)
In abstract algebra, a derivation is a function on an algebra which generalizes certain features of the derivative operator. Specifically, given an algebra A over a ring or a field K, a K-derivation is a K-linear map D: A → A that satisfies Leibniz's law: D = b + a.More...
is a linear map on a ring or algebra
Algebra over a field
In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...
which satisfies the Leibniz law (the product rule). Higher derivatives and algebraic differential operators
Algebraic differential equation
In mathematics, an algebraic differential equation is a differential equation that can be expressed by means of differential algebra. There are several such notions, according to the concept of differential algebra used....
can also be defined. They are studied in a purely algebraic setting in differential Galois theory
Differential Galois theory
In mathematics, differential Galois theory studies the Galois groups of differential equations.Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential fields, i.e. fields that are equipped with a derivation, D. Much of...
and the theory of D-module
D-module
In mathematics, a D-module is a module over a ring D of differential operators. The major interest of such D-modules is as an approach to the theory of linear partial differential equations...
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 a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
over a commutative ring R is defined by
The mapping
is then a derivation on the polynomial ringPolynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
R[X]. This definition can be extended to rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
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 algebraCommutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
, Kähler differential
Kähler differential
In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes.-Presentation:The idea was introduced by Erich Kähler in the 1930s...
s are universal derivations of a commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
or module. They can be used to define an analogue of exterior derivative
from differential geometry that applies to arbitrary algebraic varieties, instead of just smooth manifolds.
Number theory
In p-adic analysisP-adic analysis
In mathematics, p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p-adic numbers....
, the usual definition of derivative is not quite strong enough, and one requires strict differentiability
Strictly differentiable
In mathematics, strict differentiability is a modification of the usual notion of differentiability of functions that is particularly suited to p-adic analysis...
instead.
Also see arithmetic derivative
Arithmetic derivative
In number theory, the arithmetic derivative, or number derivative, is a function defined for integers, based on prime factorization, by analogy with the product rule for the derivative of a function that is used in mathematical analysis.-Definition:...
.
Type theory
Many abstract data typeAbstract data type
In computing, an abstract data type is a mathematical model for a certain class of data structures that have similar behavior; or for certain data types of one or more programming languages that have similar semantics...
s in mathematics and computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
can be described as the algebra
Universal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
generated by a transformation that maps structures based on the type back into the type. For example, the type T of binary tree
Binary tree
In computer science, a binary tree is a tree data structure in which each node has at most two child nodes, usually distinguished as "left" and "right". Nodes with children are parent nodes, and child nodes may contain references to their parents. Outside the tree, there is often a reference to...
s containing values of type A can be represented as the algebra generated by the transformation 1+A×T2→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 either way.
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 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.
This concept of a derivative of a type has practical applications, such as the zipper
Zipper (data structure)
A zipper is a technique of representing an aggregate data structure so that it is convenient for writing programs that traverse the structure arbitrarily and update its contents, especially in purely functional programming languages. The zipper was described by Gérard Huet in 1997...
technique used in functional programming languages.
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 topologyDifferential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
, a vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
may be defined as a derivation on the ring of smooth function
Smooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
s on a manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
, and a tangent vector
Tangent vector
A tangent vector is a vector that is tangent to a curve or surface at a given point.Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold....
may be defined as a derivation at a point. This allows the abstraction of the notion of a directional derivative
Directional derivative
In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P in the direction of V...
of a scalar function to general manifolds. For manifolds that are subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
s of Rn, 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 the Jacobian matrix.
On the exterior algebra
Exterior algebra
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs...
of differential forms over a smooth manifold, the exterior derivative
Exterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
is the unique linear map which satisfies a graded version of the Leibniz law and squares to zero. It is a grade 1 derivation on the exterior algebra.
The Lie derivative
Lie derivative
In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...
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 Lie bracket
Lie bracket
Lie bracket can refer to:*A bilinear binary operation defined on elements of a Lie algebra*Lie bracket of vector fields...
(vector fields form the Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
of the diffeomorphism group of the manifold). It is a grade 0 derivation on the algebra.
Together with the interior product (a degree -1 derivation on the exterior algebra defined by contraction with a vector field), the exterior derivative and the Lie derivative form a Lie superalgebra
Lie superalgebra
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2-grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry...
.
Differential geometry
In differential geometry, the covariant derivativeCovariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...
makes a choice for taking directional derivatives of vector fields along curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
s. This extends the directional derivative of scalar functions to sections of vector bundle
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s or principal bundle
Principal bundle
In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
s. In Riemannian geometry
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
, the existence of a metric chooses a unique preferred torsion
Torsion
The word torsion may refer to the following:*In geometry:** Torsion of a curve** Torsion tensor in differential geometry** The closely related concepts of Reidemeister torsion and analytic torsion ** Whitehead torsion*In algebra:** Torsion ** Tor functor* In medicine:** Ovarian...
-free covariant derivative, known as the Levi-Civita connection
Levi-Civita connection
In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...
. See also gauge covariant derivative
Gauge covariant derivative
The gauge covariant derivative is like a generalization of the covariant derivative used in general relativity. If a theory has gauge transformations, it means that some physical properties of certain equations are preserved under those transformations...
for a treatment oriented to physics.
The exterior covariant derivative
Exterior covariant derivative
In mathematics, the exterior covariant derivative, sometimes also covariant exterior derivative, is a very useful notion for calculus on manifolds, which makes it possible to simplify formulas which use a principal connection....
extends the exterior derivative to vector valued forms.
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 look locally like Banach spaceBanach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s. Thus one might want a derivative with some of the features of a functional derivative
Functional derivative
In mathematics and theoretical physics, the functional derivative is a generalization of the gradient. While the latter differentiates with respect to a vector with discrete components, the former differentiates with respect to a continuous function. Both of these can be viewed as extensions of...
and the covariant derivative
Covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...
.
The study of stochastic processes requires a form of calculus known as the Malliavin calculus
Malliavin calculus
The Malliavin calculus, named after Paul Malliavin, is a theory of variational stochastic calculus. In other words it provides the mechanics to compute derivatives of random variables....
. One notion of derivative in this setting is the H-derivative of a function on an abstract Wiener space
Abstract Wiener space
An abstract Wiener space is a mathematical object in measure theory, used to construct a "decent" measure on an infinite-dimensional vector space. It is named after the American mathematician Norbert Wiener...
.