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Gâteaux gradient

From Encyclopedia of Mathematics
Revision as of 18:52, 24 March 2012 by Ulf Rehmann (talk | contribs) (moved Gâteaux gradient to Gateaux gradient: ascii title)
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of a functional at a point of a Hilbert space

The vector in equal to the Gâteaux derivative of at . In other words, the Gâteaux gradient is defined by the formula

where as . In an -dimensional Euclidean space the Gâteaux gradient is the vector with coordinates

and is simply known as the gradient. The concept of the Gâteaux gradient may be extended to the case when is a Riemannian manifold (finite-dimensional) or an infinite-dimensional Hilbert manifold and is a smooth real function on . The growth of in the direction of its Gâteaux gradient is larger than in any other direction passing through the point .

How to Cite This Entry:
Gâteaux gradient. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=G%C3%A2teaux_gradient&oldid=12952
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article