Generalizations of the derivative

In mathematics, there are many possible generalizations of the derivative, that is, the fundamental construction of the differential calculus.

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.

One view of the derivative is that it specifies the best linear approximation to a function at a given point. Generalizing to functions from Rⁿ to R^m yields the idea of the total derivative, the best linear transformation approximating the function. Such a transformation is often represented by an n by m matrix, known as the Jacobian matrix.

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.

The determinant of the Jacobian matrix can have several natural interpretations. For functions from Rⁿ to Rⁿ, it represents the amount that the function expands or shrinks volume. If the determinant is zero, the function is said to have a singular point.

For real valued functions from Rⁿ to R, the total derivative is often called the gradient, and yields a vector each component of which is a partial derivative. This 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.

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

Convex analysis

The subderivative and subgradient are generalizations of the derivative to convex functions.

Higher order derivatives

Another simple generalization one can make to the derivative is to simply 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.

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

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

The mapping $f\mapsto f'$ is then a derivation on the polynomial ring R[X].

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 one vector field in the direction of another vector field. 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 generate a Lie superalgebra.

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.

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.

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).

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.

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.

Quantum groups

In the area of quantum groups, the q-derivative is a q-deformation of the normal derivative of a function.

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.

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