In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
is a fundamental construction of
differential calculus
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...
and admits many possible generalizations within the fields of
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
,
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
,
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
,
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, etc.
Fréchet derivative
The
Fréchet derivative
In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued fu ...
defines the derivative for general
normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
s
. Briefly, a function
,
an open subset of
, is called ''Fréchet differentiable'' at
if there exists a
bounded linear operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vect ...
such that
Functions are defined as being differentiable in some open
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...
of
, rather than at individual points, as not doing so tends to lead to many
pathological
Pathology is the study of the causal, causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when us ...
counterexamples
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is ...
.
The Fréchet derivative is quite similar to the formula for the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
found in elementary one-variable calculus,
and simply moves ''A'' to the left hand side. However, the Fréchet derivative ''A'' denotes the function
.
In
multivariable calculus
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather th ...
, in the context of differential equations defined by a vector valued function R
''n'' to R
''m'', the Fréchet derivative ''A'' is a
linear operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
on R considered as a vector space over itself, and corresponds to the ''best linear approximation'' of a function. If such an operator exists, then it is unique, and can be represented by an ''m'' by ''n''
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
known as the
Jacobian matrix
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as ...
J
''x''(ƒ) of the mapping ƒ at point ''x''. Each entry of this matrix represents a
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 (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
, 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:
J
''x''(''g
°f'') =J
ƒ(''x'')(''g'')J
''x''(ƒ). This is a higher-dimensional statement of the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
.
For real valued functions from R
''n'' to R (
scalar field
In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
s), the Fréchet derivative corresponds to a
vector field called the
total derivative
In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with resp ...
. This can be interpreted as the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
but it is more natural to use the
exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
.
The
convective derivative
Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the convecti ...
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.
For
vector-valued functions
A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could b ...
from R to R
''n'' (i.e.,
parametric curve
In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric obj ...
s), 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.
In
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, the central objects of study are
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
s, which are complex-valued functions on the
complex numbers
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
where the Fréchet derivative exists.
In
geometric calculus
In mathematics, geometric calculus extends the geometric algebra to include differentiation and integration. The formalism is powerful and can be shown to encompass other mathematical theories including differential geometry and differential f ...
, the
geometric derivative 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.
Exterior derivative and Lie derivative
On the
exterior algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
of
differential forms
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
over a
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, the
exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
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. In R
3, the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
,
curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was fi ...
, and
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
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
directional derivative
In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity s ...
s of
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
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
flux
Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ph ...
by
divergence theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
. Curl measures how much "
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
" a vector field has near a point.
The
Lie derivative
In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...
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
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identit ...
(vector fields form the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
of the
diffeomorphism group
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an Inverse function, invertible Function (mathematics), function that maps one differentiable manifold to another such that both the function and its inverse function ...
of the manifold). It is a grade 0 derivation on the algebra.
Together with the
interior product
In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, or inner derivation) is a degree −1 (anti)derivation on the exterior algebra of d ...
(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
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. In most of these theories, the ...
.
Differential topology
In
differential topology
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
, a
vector field may be defined as a derivation on the ring of
smooth function
In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
s on a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, and a
tangent vector
In mathematics, 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 R''n''. More generally, tangent vectors are eleme ...
may be defined as a derivation at a point. This allows the abstraction of the notion of a
directional derivative
In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity s ...
of a scalar function to general manifolds. For manifolds that are
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s of R
''n'', this tangent vector will agree with the
directional derivative
In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity s ...
.
The
differential or pushforward of a map between manifolds is the induced map between tangent spaces of those maps. It abstracts the
Jacobian matrix
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as ...
.
Covariant derivative
In
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, the
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 different ...
makes a choice for taking directional derivatives of vector fields along
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
s. This extends the directional derivative of scalar functions to sections of
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 (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
s or
principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equip ...
s. In
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 that varies smoothly from point to poin ...
, the existence of a metric chooses a unique preferred
torsion
Torsion may refer to:
Science
* Torsion (mechanics), the twisting of an object due to an applied torque
* Torsion of spacetime, the field used in Einstein–Cartan theory and
** Alternatives to general relativity
* Torsion angle, in chemistry
Bi ...
-free covariant derivative, known as the
Levi-Civita connection
In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves th ...
. See also
gauge covariant derivative
The gauge covariant derivative is a variation of the covariant derivative used in general relativity, quantum field theory and fluid dynamics. If a theory has gauge transformations, it means that some physical properties of certain equations are p ...
for a treatment oriented to physics.
The
exterior covariant derivative
In the mathematical field of differential geometry, the exterior covariant derivative is an extension of the notion of exterior derivative to the setting of a differentiable principal bundle or vector bundle with a connection.
Definition
Let ''G' ...
extends the exterior derivative to vector valued forms.
Weak derivatives
Given a function
which is
locally integrable In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions li ...
, but not necessarily classically differentiable, a
weak derivative
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (''strong derivative'') for functions not assumed differentiable, but only integrable, i.e., to lie in the L''p'' space L^1( ,b.
The method of ...
may be defined by means of
integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
. First define test functions, which are infinitely differentiable and compactly supported functions
, and
multi-indices
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distribution (mathematics), distributions, by generalising the concept of an integer index not ...
, which are length
lists of integers
with
. Applied to test functions,
. Then the
weak derivative of
exists if there is a function
such that for ''all'' test functions
, we have
:
If such a function exists, then
, which is unique
almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
. This definition coincides with the classical derivative for functions
, and can be extended to a type of generalized functions called
distributions, the dual space of test functions. Weak derivatives are particularly useful in the study of partial differential equations, and within parts of functional analysis.
Higher-order and fractional derivatives
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
multivariable calculus
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather th ...
. In this case, instead of repeatedly applying the derivative, one repeatedly applies
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 (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
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
In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...
. 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
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
). Nevertheless, higher derivatives have important applications to analysis of
local extrema
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ra ...
of a function at its
critical points. For an advanced application of this analysis to topology of
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s, see
Morse theory
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiabl ...
.
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 calculus
Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D
:D f(x) = \frac f(x)\,,
and of the integration ...
. The −1 order derivative corresponds to the integral, whence the term
differintegral
In fractional calculus, an area of mathematical analysis, the differintegral (sometime also called the derivigral) is a combined differentiation/ integration operator. Applied to a function ƒ, the ''q''-differintegral of ''f'', here denoted by ...
.
Quaternionic derivatives
In
quaternionic analysis
In mathematics, quaternionic analysis is the study of function (mathematics), functions with quaternions as the domain of a function, domain and/or range. Such functions can be called functions of a quaternion variable just as Function of a real va ...
, derivatives can be defined in a similar way to real and complex functions. Since the
quaternions
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
are not commutative, the limit of the difference quotient yields two different derivatives: A left derivative
: