Generalizations of the derivative

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In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, geometry, etc.

Fréchet derivative

The Fréchet derivative defines the derivative for general normed vector spaces ${\displaystyle V,W}$. Briefly, a function ${\displaystyle f:U\to W}$, where ${\displaystyle U}$ is an open subset of ${\displaystyle V}$, is called Fréchet differentiable at ${\displaystyle x\in U}$ if there exists a

${\displaystyle A:V\to W}$ such that

Functions are defined as being differentiable in some open neighbourhood of ${\displaystyle x}$, rather than at individual points, as not doing so tends to lead to many

.

The Fréchet derivative is quite similar to the formula for the derivative found in elementary one-variable calculus,

and simply moves A to the left hand side. However, the Fréchet derivative A denotes the function ${\displaystyle t\mapsto f'(x)\cdot t}$.

In

, 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.

For real valued functions from Rⁿ to R (scalar fields), the Fréchet derivative corresponds to a vector field called the total derivative. This can be interpreted as the gradient but it is more natural to use the exterior derivative.

The

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

), 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

where the Fréchet derivative exists.

In geometric calculus, 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.^[1]

Exterior derivative and Lie derivative

On the

" a vector field has near a point.

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.

Differential topology

In

.

The

Jacobian matrix

.

Covariant derivative

In differential geometry, the covariant derivative makes a choice for taking directional derivatives of vector fields along curves. This extends the directional derivative of scalar functions to sections of vector bundles or principal bundles. 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.

Weak derivatives

Given a function ${\displaystyle u:\mathbb {R} ^{n}\to \mathbb {R} }$ which is locally integrable, but not necessarily classically differentiable, a weak derivative may be defined by means of integration by parts. First define test functions, which are infinitely differentiable and compactly supported functions ${\displaystyle \varphi \in C_{c}^{\infty }\left(\mathbb {R} ^{n}\right)}$, and multi-indices, which are length ${\displaystyle n}$ lists of integers ${\displaystyle \alpha =(\alpha _{1},\dots ,\alpha _{n})}$ with ${\textstyle |\alpha |:=\sum _{1}^{n}\alpha _{i}}$. Applied to test functions, ${\textstyle D^{\alpha }\varphi :={\frac {\partial ^{|\alpha |}\varphi }{\partial x_{1}^{\alpha _{1}}\dotsm x_{n}^{\alpha _{n}}}}}$. Then the ${\textstyle \alpha ^{\text{th}}}$ weak derivative of ${\displaystyle u}$ exists if there is a function ${\displaystyle v:\mathbb {R} ^{n}\to \mathbb {R} }$ such that for all test functions ${\displaystyle \varphi }$, we have

${\displaystyle \int _{\mathbb {R} ^{n}}u\ D^{\alpha }\!\varphi \ dx=(-1)^{|\alpha |}\int _{\mathbb {R} ^{n}}v\ \varphi \ dx}$

If such a function exists, then ${\displaystyle D^{\alpha }u:=v}$, which is unique almost everywhere. This definition coincides with the classical derivative for functions ${\displaystyle u\in C^{|\alpha |}\left(\mathbb {R} ^{n}\right)}$, 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

.

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. The −1 order derivative corresponds to the integral, whence the term differintegral.

Quaternionic derivatives

In quaternionic analysis, derivatives can be defined in a similar way to real and complex functions. Since the quaternions ${\displaystyle \mathbb {H} }$ are not commutative, the limit of the difference quotient yields two different derivatives: A left derivative

${\displaystyle \lim _{h\to 0}\left[h^{-1}\left(f(a+h)-f(a)\right)\right]}$

and a right derivative

${\displaystyle \lim _{h\to 0}\left[\left(f(a+h)-f(a)\right)h^{-1}\right].}$

The existence of these limits are very restrictive conditions. For example, if ${\displaystyle f:\mathbb {H} \to \mathbb {H} }$ has left-derivatives at every point on an open connected set ${\displaystyle U\subset \mathbb {H} }$, then ${\displaystyle f(q)=a+qb}$ for ${\displaystyle a,b\in \mathbb {H} }$.

Difference operator, q-analogues and time scales

• The q-derivative of a function is defined by the formula
  $${\displaystyle D_{q}f(x)={\frac {f(qx)-f(x)}{(q-1)x}}.}$$

  
  For x nonzero, 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
  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 groups
  have changed the situation dramatically, and the popularity of q-analogues is on the rise.
• The
  difference equations
  is another discrete analog of the standard derivative.
  $${\displaystyle \Delta f(x)=f(x+1)-f(x)}$$

  
• The q-derivative, the difference operator and the standard derivative can all be viewed as the same thing on different
  time scales
  . For example, taking ${\displaystyle \varepsilon =(q-1)x}$, we may have
  $${\displaystyle {\frac {f(qx)-f(x)}{(q-1)x}}={\frac {f(x+\varepsilon )-f(x)}{\varepsilon }}.}$$

  
  The q-derivative is a special case of the Hahn difference,^[2]
  $${\displaystyle {\frac {f(qx+\omega )-f(x)}{qx+\omega -x}}.}$$

  
  The Hahn difference is not only a generalization of the q-derivative but also an extension of the forward difference.
• Also note that the q-derivative is nothing but a special case of the familiar derivative. Take ${\displaystyle z=qx}$. Then we have,
  $${\displaystyle \lim _{z\to x}{\frac {f(z)-f(x)}{z-x}}=\lim _{q\to 1}{\frac {f(qx)-f(x)}{qx-x}}=\lim _{q\to 1}{\frac {f(qx)-f(x)}{(q-1)x}}.}$$

  

Derivatives in algebra

In algebra, generalizations of the derivative can be obtained by imposing the Leibniz rule of differentiation in an algebraic structure, such as a ring or a Lie algebra.

Derivations

A

, 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

${\displaystyle \left(a_{d}x^{d}+a_{d-1}x^{d-1}+\cdots +a_{1}x+a_{0}\right)'=da_{d}x^{d-1}+(d-1)a_{d-1}x^{d-2}+\cdots +a_{1}.}$

The mapping ${\displaystyle f\mapsto f'}$ is then a derivation on the polynomial ring R[X]. This definition can be extended to rational functions 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.

Derivative of a type

In type theory, many abstract data types 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 trees 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

.

Differential operators

A differential operator combines several derivatives, possibly of different orders, in one algebraic expression. This is especially useful in considering ordinary linear differential equations with constant coefficients. For example, if f(x) is a twice differentiable function of one variable, the differential equation ${\displaystyle f''+2f'-3f=4x-1}$ may be rewritten in the form ${\displaystyle L(f)=4x-1}$, 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 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

Combining derivatives of different variables results in a notion of a

can be defined which allow for fractional calculus.

Some of these operators are so important that they have their own names:

• The Laplace operator or Laplacian on R³ is a second-order partial differential operator Δ given by the divergence of the gradient of a scalar function of three variables, or explicitly as
  $${\displaystyle \Delta ={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}.}$$

  
  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 tensor of Minkowski space, instead of the Euclidean dot product
  of R³:
  $${\displaystyle \square ={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}.}$$

  
• The
  fractional-linear map
  , in much the same way that a normal derivative describes how a function is approximated by a linear map.
• The Wirtinger derivatives are a set of differential operators that permit the construction of a differential calculus for complex functions that is entirely analogous to the ordinary differential calculus for functions of real variables.

Other generalizations

In functional analysis, 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 dimensional vector space. An important case is the variational derivative in the calculus of variations.

The

used in convex analysis.

In

algebraic varieties

, instead of just smooth manifolds.

In

strict differentiability

instead.

The

.

In

, used for changing variables, to measures. It expresses one measure μ in terms of another measure ν (under certain conditions).

The

stochastic processes

.

Laplacians and differential equations using the Laplacian can be defined on fractals. There is no completely satisfactory analog of the first-order derivative or gradient.^[3]

The Carlitz derivative is an operation similar to usual differentiation but with the usual context of real or complex numbers changed to

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

.

Multiplicative calculus replaces addition with multiplication, and hence rather than dealing with the limit of a ratio of differences, it deals with the limit of an exponentiation of ratios. This allows the development of the geometric derivative and bigeometric derivative. Moreover, just like the classical differential operator has a discrete analog, the difference operator, there are also discrete analogs of these multiplicative derivatives.

See also

• Arithmetic derivative – Function defined on integers in number theory
• Automatic differentiation – Techniques to evaluate the derivative of a function specified by a computer program
• Brzozowski derivative – Function defined on formal languages in computer science
• Dini derivative – Class of generalisations of the derivative
• Fractal derivative – Generalization of derivative to fractals
• Hasse derivative – Mathematical concept
• Logarithmic derivative – Mathematical operation in calculus
• Logarithmic differentiation – Method of mathematical differentiation
• Non-classical analysis
   – Branch of mathematicsPages displaying short descriptions of redirect targets
• Numerical differentiation – Use of numerical analysis to estimate derivatives of functions
• Pincherle derivative – Type of derivative of a linear operator
• q-derivative – Q-analog of the ordinary derivative
• Semi-differentiability
• Symmetric derivative – generalization of the derivativePages displaying wikidata descriptions as a fallback
• Topological derivative

Notes