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 $ X $ and $ Y $ be topological vector spaces. Let a mapping $ f $ be defined on an open subset $ V $ of $ X $ and let it take values in $ Y $. If the difference $ f ( x _ {0} + h ) - f ( x _ {0} ) $, where $ x _ {0} \in V $ and $ x _ {0} + h \in V $, can be approximated by a function $ l _ {x _ {0} } : X \rightarrow Y $ which is linear with respect to the increment $ h $, then $ f $ is known as a differentiable mapping at $ x _ {0} $. The approximating linear function $ l _ {x _ {0} } $ is said to be the derivative or the differential of the mapping at $ x _ {0} $ and is denoted by the symbol $ f ^ { \prime } ( x _ {0} ) $ or $ df ( x _ {0} ) $. 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 $ h \in X $, denoted by the symbol $ f ^ { \prime } ( x _ {0} ) h $, $ d f ( x _ {0} ) ( h) $ or $ d _ {h} f ( x _ {0} ) $, is known as the differential of the mapping $ f $ at the point $ x _ {0} $ for the increment $ h $.

Depending on the meaning attributed to the approximation of the increment $ f ( x _ {0} + h ) - f ( x _ {0} ) $ by a linear expression in $ h $, there result different concepts of differentiability and of the derivative. For the most important existing definitions see [1], .

Let $ F $ be the set of all mappings from $ X $ into $ Y $ and let $ \tau $ be some topology or pseudo-topology in $ F $. A mapping $ r \in F $ is small at zero if the curve

$$ r _ {t} : \frac{r ( t x ) }{t} , $$

conceived of as a mapping

$$ t \rightarrow \left [ x \rightarrow \frac{r ( t x ) }{t} \right ] $$

of the straight line $ - \infty < t < \infty $ into $ F $, is continuous at zero in the (pseudo-) topology $ \tau $. Now, a mapping $ f \in F $ is differentiable at a point $ x _ {0} $ if there exists a continuous linear mapping $ l _ {x _ {0} } $ such that the mapping

$$ r : h \rightarrow f ( x _ {0} + h ) - f ( x _ {0} ) - l _ {x _ {0} } ( h) $$

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

If $ X = \mathbf R ^ {n} $ and $ Y = \mathbf R ^ {m} $, the derivative $ f ^ { \prime } ( x _ {0} ) $ of a differentiable mapping $ f ( x) = ( f _ {1} ( x) \dots f _ {m} ( x) ) $, where $ x = ( x _ {1} \dots x _ {n} ) $, is defined by the Jacobi matrix $ \| \partial f _ {i} ( x _ {0} ) / \partial x _ {j} \| $, and is a continuous linear mapping from $ \mathbf R ^ {n} $ into $ \mathbf R ^ {m} $.

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:

$$ ( f + g ) ^ \prime ( x _ {0} ) = f ^ { \prime } ( x _ {0} ) + g ^ \prime ( x _ {0} ) , $$

$$ ( \alpha f ) ^ \prime ( x _ {0} ) = \alpha f ^ { \prime } ( x _ {0} ) ; $$

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

$$ ( f \circ g ) ^ \prime ( x _ {0} ) = f ^ { \prime } ( g ( x _ {0} ) ) \circ g ^ \prime ( x _ {0} ) , $$

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 $ X $ and $ Y $ 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)