Differentiation of a mapping

Finding the differential or, in other words, the principal linear part (of increment) of the mapping. The finding of the differential, i.e. the approximation of the mapping in a neighbourhood of some point by linear mappings, is a highly important operation in differential calculus. A very general framework for differential calculus can be formulated in the setting of topological vector spaces.

Let and be topological vector spaces. Let a mapping be defined on an open subset of and let it take values in . If the difference , where and , can be approximated by a function which is linear with respect to the increment , then is known as a differentiable mapping at . The approximating linear function is said to be the derivative or the differential of the mapping at and is denoted by the symbol or . Mappings with identical derivatives at a given point are said to be mutually tangent mappings at this point. The value of the approximating function on an element , denoted by the symbol , or , is known as the differential of the mapping at the point for the increment .

Depending on the meaning attributed to the approximation of the increment by a linear expression in , there result different concepts of differentiability and of the derivative. For the most important existing definitions see [1], .

Let be the set of all mappings from into and let be some topology or pseudo-topology in . A mapping is small at zero if the curve

conceived of as a mapping

of the straight line into , is continuous at zero in the (pseudo-) topology . Now, a mapping is differentiable at a point if there exists a continuous linear mapping such that the mapping

is small at zero. Depending on the choice of in various definitions of derivatives are obtained. Thus, if the topology of pointwise convergence is selected for , one obtains differentiability according to Gâteaux (cf. Gâteaux derivative). If and are Banach spaces and the topology in is the topology of uniform convergence on bounded sets in , one obtains differentiability according to Fréchet (cf. Fréchet derivative).

If and , the derivative of a differentiable mapping , where , is defined by the Jacobi matrix , and is a continuous linear mapping from into .

Derivatives of mappings display many of the properties of the derivatives of functions of one variable. For instance, under very general assumptions, they display the property of linearity:

and in many cases the formula for differentiation of composite functions:

is applicable; the generalized mean-value theorem of Lagrange is valid for mappings into locally convex spaces.

The concept of a differentiable mapping is extended to the case when and are smooth differentiable manifolds, both finite-dimensional and infinite-dimensional [4], [5], [6]. Differentiable mappings between infinite-dimensional spaces and their derivatives were defined for the first time by V. Volterra (1887), M. Fréchet (1911) and R. Gâteaux (1913). For a more detailed history of the development of the concept of a derivative in higher-dimensional spaces see .

References

[1]	A. Frölicher, W. Bucher, "Calculus in vector spaces without norm" , Lect. notes in math. , 30 , Springer (1966)
[2a]	V.I. Averbukh, O.G. Smolyanov, "The theory of differentiation in linear topological spaces" Russian Math. Surveys , 22 : 6 (1967) pp. 201–258 Uspekhi Mat. Nauk , 22 : 6 (1967) pp. 201–260
[2b]	V.I. Averbukh, O.G. Smolyanov, "The various definitions of the derivative in linear topological spaces" Russian Math. Surveys , 23 : 4 (1968) pp. 67–113 Uspekhi Mat. Nauk , 23 : 4 (1968) pp. 67–116
[3]	J.A. Dieudonné, "Foundations of modern analysis" , Acad. Press (1961) (Translated from French)
[4]	S. Lang, "Introduction to differentiable manifolds" , Interscience (1967) pp. App. III
[5]	N. Bourbaki, "Elements of mathematics. Differentiable and analytic manifolds" , Addison-Wesley (1966) (Translated from French)
[6]	M. Spivak, "Calculus on manifolds" , Benjamin (1965)