Derivative (generalizations)
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In mathematics, there are many possible generalizations of the derivative, that is, the fundamental construction of the differential calculus.
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[edit] Multivariable calculus
The derivative is often met for the first time as an operation on a single real function of a single real variable. One of the simplest settings for generalizations is to vector valued functions of several variables (most often the domain forms a vector space as well). This is the field of multivariable calculus.
In one-variable calculus, we say that a function is differentiable at a point x if the limit
- <math>\lim_{h \to 0}\frac{f(x+h) - f(x)}{h}</math>
exists, its value is then the derivative, <math>f'(x)</math>. A function is differentiable on an interval if it is differentiable at every point within the interval.
We can generalize to functions mapping Rm to Rn as follows: <math>f</math> is differentiable at <math>x</math> if there exists a linear operator <math>A(x)</math> (depending on <math>x</math>) such that
- <math>\lim_{||h|| \to 0}\frac{||f(x+h) - f(x) - A(x)h||}{||h||} = 0</math>.
Note that, in general, we concern ourselves mostly with functions being differentiable in some open neighbourhood of <math>x</math> rather than at individual points, as not doing so tends to lead to manifold pathological counterexamples. If in turn a function <math>g</math> from Rn to Rp is differentiable in some neighbourhood of <math> f(x) </math> with derivative <math>B(f(x))</math> then the chain rule in this case states that derivative of the composition of functions <math>g(f(x))</math> is a composition of the corresponding linear operators <math> B(f(x))(A(x)) .</math>
An m by n matrix, of the linear operator <math>A(x)</math> is known as Jacobian matrix Jx(f) of the mapping <math>f</math> at point <math>x</math>. 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 <math>gf</math> is a product of corresponding Jacobian matrices: Jx(gf) =Jx(f)Jf(x)(g).
For real valued functions from Rn to R (scalar fields), 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 derivatives 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 (a.k.a. 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 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.
[edit] Convex analysis
The subderivative and subgradient are generalizations of the derivative to convex functions.
[edit] Derivatives of non-unitary order
Another simple generalization one can make to the derivative is to apply it more than once, obtaining second order derivative (and higher), as defined in the article on derivatives. This notion can be generalized.
In addition to n-th derivatives for any natural number n, using various methods, one can take derivatives to fractional or negative powers. The -1 order derivative will then correspond to the integral, whence the term differintegral. The study of different possible definitions and notions of derivatives to nonnatural numbered powers is known as fractional calculus.
In multivariate calculus, the second order derivative of a scalar function is given by the Hessian matrix, which is the matrix of second order partial derivatives. It is used in finding local extrema, and also in Morse theory.
The Laplacian is a second-order differential operator given by the divergence of the gradient of a scalar function on Rn. The definition of the d'Alembertian is similar to the Laplacian's, but it uses the indefinite metric of Minkowski space, instead of the Euclidean dot product of Rn.
[edit] Algebra
A derivation is a linear map on a ring or algebra which satisfies the Leibniz law (product rule). They are studied in a purely algebraic setting in differential Galois theory, but also turn up in many other areas, where they often agree with less algebraic definitions of derivatives.
In particular, the formal derivative of a polynomial over a commutative ring R is defined by
- <math>(a_dx^d + a_{d-1}x^{d-1} + \cdots+a_1x+a_0)' = da_dx^{d-1}+(d-1)a_{d-1}x^{d-2} + \cdots+a_1.</math>
The mapping <math>f\mapsto f'</math> is then a derivation on the polynomial ring R[X]. This definition can be extended to rational functions as well.
Also see Pincherle derivative.
[edit] Type theory
Zero, addition, product, power — these concepts known from the area of arithmetic have their analogies also in set theory, category theory, type theory. Here, for example, let us see some set theoretical ones:
- <math>\empty</math>
- <math>A \times B</math>
- <math>A^n</math> for each <math>n\in\mathbb N</math> (using n-tuples)
- <math>B^A</math>, sometimes writen as <math>A \to B</math>
Also
- <math>A + B</math>
can be defined for sets as a fruitful concept. It something similar to the disjoint union of sets, but it uses labels to achieve a partition-like construct <ref>More precisely, it is a two-arguments case of the more general construct
- <math>\sum_\Lambda \vec A = \left\{\left\langle\lambda, a\right\rangle \in \Lambda \times \bigcup\mathcal A \mid \lambda\in\Lambda \land a \in \vec A_\lambda\right\}</math> where <math>\vec A : \Lambda \to \mathcal A</math></ref>.
There are analogous constructs for types, too (see also typeful functional programming languages). Now let us see parametric types, e.g.
- <math>F(X) = X^3</math>
Thus, we can write “polynomials” for types. Let us define the derivative here as we define it for polynomials over a ring!
- <math>F^\prime(X) = X^2 + X^2 + X^2</math>
It can represent a (homogenous) triple “with a hole”.
There are also more interesting constructs, than such polynomial ones. This notion of “derivative” can be extended also to them. The question arises now, if this concept is fruitful. Yes, it is: it has practical applications (in functional programming): see Zipper (data structure).
A good starting point for the topic: Infinitesimal types, it provides both an informal introduction and links to exact details.
[edit] Differential topology
In differential topology, a vector field may be defined as a derivation on the ring of smooth functions 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 subsets of Rn, this tangent vector will agree with the directional derivative defined above.
The 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.
The inner derivative is a grade –1 derivation on the exterior algebra of forms. Together, the exterior derivative, the Lie derivative, and the inner derivative span a Lie superalgebra.
[edit] Differential geometry
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.
[edit] Complex analysis
In complex analysis, the central objects of study are holomorphic functions, 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.
[edit] Functional analysis
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.
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 spaces.
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 spaces, 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 operators can be defined which allow for fractional calculus.
[edit] Algebraic geometry
In algebraic geometry, the Kähler differential allows the definition of the exterior derivative to be extended to arbitrary algebraic varieties, instead of just smooth manifolds.
[edit] Number theory
In p-adic analysis, the usual definition of derivative is not quite strong enough, and one requires strict differentiability instead.
[edit] Quantum groups
In the area of quantum groups, the q-derivative is a q-deformation of the normal derivative of a function.
[edit] 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 spaces. 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.
[edit] Notes
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