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
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conceived of as a mapping
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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
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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:
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and in many cases the formula for differentiation of composite functions:
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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) |
Differentiation of a mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Differentiation_of_a_mapping&oldid=46698