Difference between revisions of "Gâteaux gradient"
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Revision as of 07:54, 26 March 2012
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=23304
Gâteaux gradient. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=G%C3%A2teaux_gradient&oldid=23304
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article