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''Volterra derivative''
 
''Volterra derivative''
  
One of the first concepts of a derivative in an infinite-dimensional space. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f0420401.png" /> be some functional of a continuous function of one variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f0420402.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f0420403.png" /> be some interior point of the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f0420404.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f0420405.png" />, where the variation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f0420406.png" /> is different from zero in a small neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f0420407.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f0420408.png" />; and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f0420409.png" />. The limit
+
One of the first concepts of a derivative in an infinite-dimensional space. Let $  I ( y) $
 +
be some functional of a continuous function of one variable $  y ( x) $;  
 +
let $  x _ {0} $
 +
be some interior point of the segment $  [ x _ {1} , x _ {2} ] $;  
 +
let $  y _ {1} ( x) = y _ {0} ( x) + \delta y ( x) $,  
 +
where the variation $  \delta y ( x) $
 +
is different from zero in a small neighbourhood $  [ a, b] $
 +
of $  x _ {0} $;  
 +
and let $  \sigma = \int _ {a}  ^ {b} \delta y ( x)  dx $.  
 +
The limit
  
<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/f/f042/f042040/f04204010.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {\begin{array}{c}
 +
\sigma \rightarrow 0, \\
 +
a , b \rightarrow x _ {0}
 +
\end{array}
 +
} \
  
assuming that it exists, is called the functional derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f04204011.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f04204012.png" />. For example, for the simplest functional of the classical calculus of variations,
+
\frac{I ( y _ {1} ) - I ( y _ {0} ) } \sigma
 +
,
 +
$$
  
<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/f/f042/f042040/f04204013.png" /></td> </tr></table>
+
assuming that it exists, is called the functional derivative of  $  I $
 +
and is denoted by  $  ( \delta I ( y _ {0} )/ \delta y) \mid  _ {x = x _ {0}  } $.
 +
For example, for the simplest functional of the classical calculus of variations,
 +
 
 +
$$
 +
I ( y)  = \
 +
\int\limits _ { x _ {1} } ^ { {x _ 2 } }
 +
F ( x, y, \dot{y} )  dx,
 +
$$
  
 
the functional derivative has the form
 
the functional derivative has the form
  
<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/f/f042/f042040/f04204014.png" /></td> </tr></table>
+
$$
 +
\left .
 +
\frac{\delta I ( y _ {0} ) }{\delta y }
  
<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/f/f042/f042040/f04204015.png" /></td> </tr></table>
+
\right | _ {x = x _ {0}  } =
 +
$$
  
that is, it is the left-hand side of the Euler equation, which is a necessary condition for a minimum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f04204016.png" />.
+
$$
 +
= \
  
In theoretical questions the concept of a functional derivative has only historical interest, and in practice has been supplanted by the concepts of the [[Gâteaux derivative|Gâteaux derivative]] and the [[Fréchet derivative|Fréchet derivative]]. But the concept of a functional derivative has been applied with success in numerical methods of the classical [[calculus of variations]] (see [[Variational calculus, numerical methods of|Variational calculus, numerical methods of]]).
+
\frac{\partial  F ( x _ {0} , y _ {0} ( x _ {0} ), \dot{y} _ {0} ( x _ {0} )) }{\partial  y }
 +
- {
 +
\frac{d}{dx}
 +
}
 +
\frac{\partial  F ( x _ {0} , y ( x _ {0} ),
 +
\dot{y} ( x _ {0} )) }{\partial  \dot{y} }
 +
,
 +
$$
  
 +
that is, it is the left-hand side of the Euler equation, which is a necessary condition for a minimum of  $  I ( y) $.
  
 +
In theoretical questions the concept of a functional derivative has only historical interest, and in practice has been supplanted by the concepts of the [[Gâteaux derivative|Gâteaux derivative]] and the [[Fréchet derivative|Fréchet derivative]]. But the concept of a functional derivative has been applied with success in numerical methods of the classical [[calculus of variations]] (see [[Variational calculus, numerical methods of|Variational calculus, numerical methods of]]).
  
 
====Comments====
 
====Comments====
The existence of the functional derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f04204017.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f04204018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f04204019.png" /> apparently means that the Fréchet derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f04204020.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f04204021.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f04204022.png" />, which is a continuous linear form on the space of admissible infinitesimal variations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f04204023.png" />, is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f04204024.png" /> for some continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f04204025.png" />, so that it can be continuously extended to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f04204026.png" /> the [[Delta-function|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f04204027.png" />-function]] at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f04204028.png" />. In the example this happens only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f04204029.png" /> is twice continuously differentiable.
+
The existence of the functional derivative of $  I $
 +
at $  y = y _ {0} $
 +
and $  x = x _ {0} $
 +
apparently means that the Fréchet derivative $  dI $
 +
of $  I $
 +
at $  y = y _ {0} $,  
 +
which is a continuous linear form on the space of admissible infinitesimal variations $  z $,  
 +
is of the form $  \int u ( x) \cdot z ( x)  dx $
 +
for some continuous function $  u $,  
 +
so that it can be continuously extended to $  z = $
 +
the [[Delta-function| $  \delta $-
 +
function]] at $  x = x _ {0} $.  
 +
In the example this happens only if $  y _ {0} $
 +
is twice continuously differentiable.

Latest revision as of 19:40, 5 June 2020


Volterra derivative

One of the first concepts of a derivative in an infinite-dimensional space. Let $ I ( y) $ be some functional of a continuous function of one variable $ y ( x) $; let $ x _ {0} $ be some interior point of the segment $ [ x _ {1} , x _ {2} ] $; let $ y _ {1} ( x) = y _ {0} ( x) + \delta y ( x) $, where the variation $ \delta y ( x) $ is different from zero in a small neighbourhood $ [ a, b] $ of $ x _ {0} $; and let $ \sigma = \int _ {a} ^ {b} \delta y ( x) dx $. The limit

$$ \lim\limits _ {\begin{array}{c} \sigma \rightarrow 0, \\ a , b \rightarrow x _ {0} \end{array} } \ \frac{I ( y _ {1} ) - I ( y _ {0} ) } \sigma , $$

assuming that it exists, is called the functional derivative of $ I $ and is denoted by $ ( \delta I ( y _ {0} )/ \delta y) \mid _ {x = x _ {0} } $. For example, for the simplest functional of the classical calculus of variations,

$$ I ( y) = \ \int\limits _ { x _ {1} } ^ { {x _ 2 } } F ( x, y, \dot{y} ) dx, $$

the functional derivative has the form

$$ \left . \frac{\delta I ( y _ {0} ) }{\delta y } \right | _ {x = x _ {0} } = $$

$$ = \ \frac{\partial F ( x _ {0} , y _ {0} ( x _ {0} ), \dot{y} _ {0} ( x _ {0} )) }{\partial y } - { \frac{d}{dx} } \frac{\partial F ( x _ {0} , y ( x _ {0} ), \dot{y} ( x _ {0} )) }{\partial \dot{y} } , $$

that is, it is the left-hand side of the Euler equation, which is a necessary condition for a minimum of $ I ( y) $.

In theoretical questions the concept of a functional derivative has only historical interest, and in practice has been supplanted by the concepts of the Gâteaux derivative and the Fréchet derivative. But the concept of a functional derivative has been applied with success in numerical methods of the classical calculus of variations (see Variational calculus, numerical methods of).

Comments

The existence of the functional derivative of $ I $ at $ y = y _ {0} $ and $ x = x _ {0} $ apparently means that the Fréchet derivative $ dI $ of $ I $ at $ y = y _ {0} $, which is a continuous linear form on the space of admissible infinitesimal variations $ z $, is of the form $ \int u ( x) \cdot z ( x) dx $ for some continuous function $ u $, so that it can be continuously extended to $ z = $ the $ \delta $- function at $ x = x _ {0} $. In the example this happens only if $ y _ {0} $ is twice continuously differentiable.

How to Cite This Entry:
Functional derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Functional_derivative&oldid=47013
This article was adapted from an original article by I.B. Vapnyarskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article