View source for 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d0323301.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d0323302.png" /> be topological vector spaces. Let a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d0323303.png" /> be defined on an open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d0323304.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d0323305.png" /> and let it take values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d0323306.png" />. If the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d0323307.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d0323308.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d0323309.png" />, can be approximated by a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233010.png" /> which is linear with respect to the increment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233011.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233012.png" /> is known as a differentiable mapping at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233013.png" />. The approximating linear function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233014.png" /> is said to be the derivative or the differential of the mapping at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233015.png" /> and is denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233016.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233017.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233018.png" />, denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233020.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233021.png" />, is known as the differential of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233022.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233023.png" /> for the increment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233024.png" />.

Depending on the meaning attributed to the approximation of the increment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233025.png" /> by a linear expression in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233026.png" />, there result different concepts of differentiability and of the derivative. For the most important existing definitions see [[#References|[1]]], .

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233027.png" /> be the set of all mappings from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233028.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233029.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233030.png" /> be some topology or pseudo-topology in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233031.png" />. A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233032.png" /> is small at zero if the curve

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233033.png" /></td> </tr></table>

conceived of as a mapping

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of the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233035.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233036.png" />, is continuous at zero in the (pseudo-) topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233037.png" />. Now, a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233038.png" /> is differentiable at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233039.png" /> if there exists a continuous linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233040.png" /> such that the mapping

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is small at zero. Depending on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233042.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233043.png" /> various definitions of derivatives are obtained. Thus, if the topology of pointwise convergence is selected for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233044.png" />, one obtains differentiability according to Gâteaux (cf. [[Gâteaux derivative|Gâteaux derivative]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233046.png" /> are Banach spaces and the topology in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233047.png" /> is the topology of uniform convergence on bounded sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233048.png" />, one obtains differentiability according to Fréchet (cf. [[Fréchet derivative|Fréchet derivative]]).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233050.png" />, the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233051.png" /> of a differentiable mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233052.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233053.png" />, is defined by the [[Jacobi matrix|Jacobi matrix]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233054.png" />, and is a continuous linear mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233055.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233056.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032330/d03233061.png" /> are smooth differentiable manifolds, both finite-dimensional and infinite-dimensional [[#References|[4]]], [[#References|[5]]], [[#References|[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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Frölicher,   W. Bucher,   "Calculus in vector spaces without norm" , ''Lect. notes in math.'' , '''30''' , Springer  (1966)</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.A. Dieudonné,   "Foundations of modern analysis" , Acad. Press  (1961)  (Translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Lang,   "Introduction to differentiable manifolds" , Interscience  (1967)  pp. App. III</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Differentiable and analytic manifolds" , Addison-Wesley  (1966)  (Translated from French)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  M. Spivak,   "Calculus on manifolds" , Benjamin  (1965)</TD></TR></table>