# 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)
How to Cite This Entry:
Differentiation of a mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Differentiation_of_a_mapping&oldid=46698
This article was adapted from an original article by O.G. SmolyanovV.I. SobolevV.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article