Namespaces
Variants
Actions

Difference between revisions of "Gradient"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (MR/ZBL numbers added)
m (tex encoded by computer)
 
Line 1: Line 1:
 +
<!--
 +
g0446801.png
 +
$#A+1 = 87 n = 0
 +
$#C+1 = 87 : ~/encyclopedia/old_files/data/G044/G.0404680 Gradient
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
One of the fundamental concepts in vector analysis and the theory of non-linear mappings.
 
One of the fundamental concepts in vector analysis and the theory of non-linear mappings.
  
The gradient of a scalar function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g0446801.png" /> of a vector argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g0446802.png" /> from a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g0446803.png" /> is the derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g0446804.png" /> with respect to the vector argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g0446805.png" />, i.e. the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g0446806.png" />-dimensional vector with components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g0446807.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g0446808.png" />. The following notations exist for the gradient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g0446809.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468010.png" />:
+
The gradient of a scalar function $  f $
 +
of a vector argument $  t = ( t  ^ {1} \dots t  ^ {n} ) $
 +
from a Euclidean space $  E  ^ {n} $
 +
is the derivative of $  f $
 +
with respect to the vector argument $  t $,  
 +
i.e. the $  n $-
 +
dimensional vector with components $  \partial  f / \partial  t  ^ {i} $,  
 +
$  1 \leq  i \leq  n $.  
 +
The following notations exist for the gradient of $  f $
 +
at  $  t _ {0} $:
 +
 
 +
$$
 +
\mathop{\rm grad}  f ( t _ {0} ),\ \
 +
\nabla f ( t _ {0} ),\ \
 +
 
 +
\frac{\partial  f ( t _ {0} ) }{\partial  t }
 +
,\ \
 +
f ^ { \prime } ( t _ {0} ) ,\ \
 +
\left .  
 +
\frac{\partial  f }{\partial  t }
 +
\right | _ {t _ {0}  } .
 +
$$
 +
 
 +
The gradient is a [[Covariant vector|covariant vector]]: the components of the gradient, computed in two different coordinate systems  $  t = ( t  ^ {1} \dots t  ^ {n} ) $
 +
and  $  \tau = ( \tau  ^ {1} \dots \tau  ^ {n} ) $,
 +
are connected by the relations:
 +
 
 +
$$
 +
 
 +
\frac{\partial  f }{\partial  t  ^ {i} }
 +
 
 +
( \tau ( t))  = \
 +
\sum _ {j = 1 } ^ { n }
 +
 
 +
\frac{\partial  f ( \tau ) }{\partial  \tau  ^ {j} }
 +
\
 +
 
 +
\frac{\partial  \tau  ^ {j} }{\partial  t  ^ {i} }
 +
.
 +
$$
  
<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/g/g044/g044680/g04468011.png" /></td> </tr></table>
+
The vector  $  f ^ { \prime } ( t _ {0} ) $,
 +
with its origin at  $  t _ {0} $,
 +
points to the direction of fastest increase of  $  f $,
 +
and is orthogonal to the level lines or surfaces of  $  f $
 +
passing through  $  t _ {0} $.
  
The gradient is a [[Covariant vector|covariant vector]]: the components of the gradient, computed in two different coordinate systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468013.png" />, are connected by the relations:
+
The derivative of the function at  $  t _ {0} $
 +
in the direction of an arbitrary unit vector  $  \mathbf N = ( N  ^ {1} \dots N  ^ {n} ) $
 +
is equal to the projection of the gradient function onto this direction:
  
<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/g/g044/g044680/g04468014.png" /></td> </tr></table>
+
$$ \tag{1 }
  
The vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468015.png" />, with its origin at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468016.png" />, points to the direction of fastest increase of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468017.png" />, and is orthogonal to the level lines or surfaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468018.png" /> passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468019.png" />.
+
\frac{\partial  f ( t _ {0} ) }{\partial  \mathbf N }
 +
  = \
 +
( f ^ { \prime } ( t _ {0} ), \mathbf N )  \equiv \
 +
\sum _ {j = 1 } ^ { n }
  
The derivative of the function at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468020.png" /> in the direction of an arbitrary unit vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468021.png" /> is equal to the projection of the gradient function onto this direction:
+
\frac{\partial  f ( t _ {0} ) }{\partial  t  ^ {j} }
  
<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/g/g044/g044680/g04468022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
N  ^ {j}  = | f ^ { \prime } ( t _ {0} ) |  \cos  \phi ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468023.png" /> is the angle between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468025.png" />. The maximal directional derivative is attained if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468026.png" />, i.e. in the direction of the gradient, and that maximum is equal to the length of the gradient.
+
where $  \phi $
 +
is the angle between $  \mathbf N $
 +
and $  f ^ { \prime } ( t _ {0} ) $.  
 +
The maximal directional derivative is attained if $  \phi = 0 $,  
 +
i.e. in the direction of the gradient, and that maximum is equal to the length of the gradient.
  
The concept of a gradient is closely connected with the concept of the [[Differential|differential]] of a function. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468027.png" /> is differentiable at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468028.png" />, then, in a neighbourhood of that point,
+
The concept of a gradient is closely connected with the concept of the [[Differential|differential]] of a function. If $  f $
 +
is differentiable at $  t _ {0} $,  
 +
then, in a neighbourhood of that point,
  
<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/g/g044/g044680/g04468029.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
f ( t)  = f ( t _ {0} ) + ( f ^ { \prime } ( t _ {0} ),\
 +
t - t _ {0} ) + o ( | t - t _ {0} | ),
 +
$$
  
i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468030.png" />. The existence of the gradient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468031.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468032.png" /> is not sufficient for formula (2) to be valid.
+
i.e. $  df = ( f ^ { \prime } ( t _ {0} ), dt) $.  
 +
The existence of the gradient of $  f $
 +
at $  t _ {0} $
 +
is not sufficient for formula (2) to be valid.
  
A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468033.png" /> at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468034.png" /> is called a stationary (critical or extremal) point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468035.png" />. An example of such a point is a local extremal point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468036.png" />, and the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468038.png" />, is employed to find an extremal point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468039.png" />.
+
A point $  t _ {0} $
 +
at which $  f ^ { \prime } ( t _ {0} ) = 0 $
 +
is called a stationary (critical or extremal) point of $  f $.  
 +
An example of such a point is a local extremal point of $  f $,  
 +
and the system $  \partial  f ( t _ {0} ) / \partial  t  ^ {i} = 0 $,
 +
$  1 \leq  i \leq  n $,  
 +
is employed to find an extremal point $  t _ {0} $.
  
 
The following formulas can be used to compute the value of the gradient:
 
The following formulas can be used to compute the value of the gradient:
  
<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/g/g044/g044680/g04468040.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm grad}  ( \lambda f  )  = \
 +
\lambda  \mathop{\rm grad}  f,\ \
 +
\lambda = \textrm{ const } ,
 +
$$
 +
 
 +
$$
 +
\mathop{\rm grad}  ( f + g)  =   \mathop{\rm grad}  f +  \mathop{\rm grad}  g,
 +
$$
  
<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/g/g044/g044680/g04468041.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm grad}  ( fg)  = g  \mathop{\rm grad}  f + f  \mathop{\rm grad}  g,
 +
$$
  
<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/g/g044/g044680/g04468042.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm grad} \left ( {
 +
\frac{f}{g}
 +
}  \right )  =
 +
\frac{1}{g  ^ {2} }
 +
( g  \mathop{\rm grad}  f - f  \mathop{\rm grad}  g).
 +
$$
  
<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/g/g044/g044680/g04468043.png" /></td> </tr></table>
+
The gradient  $  f ^ { \prime } ( t _ {0} ) $
 +
is the derivative at  $  t _ {0} $
 +
with respect to volume of the vector function given by
  
The gradient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468044.png" /> is the derivative at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468045.png" /> with respect to volume of the vector function given by
+
$$
 +
\Phi ( E)  = \
 +
\int\limits _ {t \in \partial  E } f ( t) \mathbf M  ds,
 +
$$
  
<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/g/g044/g044680/g04468046.png" /></td> </tr></table>
+
where  $  E $
 +
is a domain with boundary  $  \partial  E $,
 +
$  t _ {0} \in E $,
 +
$  ds $
 +
is the area element of  $  \partial  E $,
 +
and  $  \mathbf M $
 +
is the unit vector of the outward normal to  $  \partial  E $.  
 +
In other words,
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468047.png" /> is a domain with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468050.png" /> is the area element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468051.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468052.png" /> is the unit vector of the outward normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468053.png" />. In other words,
+
$$
 +
f ^ { \prime } ( t _ {0} )  = \
 +
\lim\limits 
 +
\frac{\Phi ( E) }{ \mathop{\rm vol}  E }
  
<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/g/g044/g044680/g04468054.png" /></td> </tr></table>
+
\  \textrm{ as } \ \
 +
E \rightarrow t _ {0} .
 +
$$
  
 
Formulas (1), (2) and the properties of the gradient listed above indicate that the concept of a gradient is invariant with respect to the choice of a coordinate system.
 
Formulas (1), (2) and the properties of the gradient listed above indicate that the concept of a gradient is invariant with respect to the choice of a coordinate system.
  
In a curvilinear coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468055.png" />, in which the square of the linear element is
+
In a curvilinear coordinate system $  x = ( x  ^ {1} \dots x  ^ {n} ) $,  
 +
in which the square of the linear element is
 +
 
 +
$$
 +
ds  ^ {2}  = \
 +
\sum _ {i, j = 1 } ^ { n }
 +
g _ {ij} ( x)  dx  ^ {i}  dx  ^ {j} ,
 +
$$
 +
 
 +
the components of the gradient of  $  f $
 +
with respect to the unit vectors tangent to coordinate lines at  $  x $
 +
are
  
<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/g/g044/g044680/g04468056.png" /></td> </tr></table>
+
$$
 +
\sum _ {j = 1 } ^ { n }
 +
g ^ {ij} ( x)
  
the components of the gradient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468057.png" /> with respect to the unit vectors tangent to coordinate lines at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468058.png" /> are
+
\frac{\partial  f }{\partial  x  ^ {j} }
 +
,\ \
 +
1 \leq  i \leq  n,
 +
$$
  
<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/g/g044/g044680/g04468059.png" /></td> </tr></table>
+
where the matrix  $  \| g  ^ {ij} \| $
 +
is the inverse of the matrix  $  \| g _ {ij} \| $.
  
where the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468060.png" /> is the inverse of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468061.png" />.
+
The concept of a gradient for more general vector functions of a vector argument is introduced by means of equation (2). Thus, the gradient is a linear operator the effect of which on the increment  $  t - t _ {0} $
 +
of the argument is to yield the principal linear part of the increment  $  f( t) - f( t _ {0} ) $
 +
of the vector function  $  f $.  
 +
E.g., if  $  f = ( f ^ { 1 } \dots f ^ { m } ) $
 +
is an  $  m $-
 +
dimensional vector function of the argument  $  t = ( t  ^ {1} \dots t  ^ {n} ) $,
 +
then its gradient at a point  $  t _ {0} $
 +
is the [[Jacobi matrix|Jacobi matrix]]  $  J = J ( t _ {0} ) $
 +
with components  $  ( \partial  f ^ { i } / \partial  t  ^ {j} ) ( t _ {0} ) $,
 +
$  1 \leq  i \leq  m $,
 +
$  1 \leq  j \leq  n $,
 +
and
  
The concept of a gradient for more general vector functions of a vector argument is introduced by means of equation (2). Thus, the gradient is a linear operator the effect of which on the increment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468062.png" /> of the argument is to yield the principal linear part of the increment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468063.png" /> of the vector function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468064.png" />. E.g., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468065.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468066.png" />-dimensional vector function of the argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468067.png" />, then its gradient at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468068.png" /> is the [[Jacobi matrix|Jacobi matrix]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468069.png" /> with components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468070.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468071.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468072.png" />, and
+
$$
 +
f ( t) = f ( t _ {0} ) +
 +
J ( t - t _ {0} ) + o ( t - t _ {0} ),
 +
$$
  
<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/g/g044/g044680/g04468073.png" /></td> </tr></table>
+
where  $  o ( t - t _ {0} ) $
 +
is an  $  m $-
 +
dimensional vector of length  $  o ( | t - t _ {0} | ) $.  
 +
The matrix  $  J $
 +
is defined by the limit transition
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468074.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468075.png" />-dimensional vector of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468076.png" />. The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468077.png" /> is defined by the limit transition
+
$$ \tag{3 }
 +
\lim\limits _ {\rho \rightarrow 0 } \
  
<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/g/g044/g044680/g04468078.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
\frac{f ( t _ {0} + \rho \tau ) - f ( t _ {0} ) } \rho
 +
  =  J \tau ,
 +
$$
  
for any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468079.png" />-dimensional vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468080.png" />.
+
for any fixed $  n $-
 +
dimensional vector $  \tau $.
  
 
In an infinite-dimensional Hilbert space definition (3) is equivalent to the definition of differentiability according to Fréchet, the gradient then being identical with the [[Fréchet derivative|Fréchet derivative]].
 
In an infinite-dimensional Hilbert space definition (3) is equivalent to the definition of differentiability according to Fréchet, the gradient then being identical with the [[Fréchet derivative|Fréchet derivative]].
  
If the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468081.png" /> lie in an infinite-dimensional vector space, various types of limit transitions in (3) are possible (see, for example, [[Gâteaux derivative|Gâteaux derivative]]).
+
If the values of $  f $
 +
lie in an infinite-dimensional vector space, various types of limit transitions in (3) are possible (see, for example, [[Gâteaux derivative|Gâteaux derivative]]).
  
In the theory of tensor fields on a domain of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468082.png" />-dimensional affine space with a connection, the gradient serves to describe the principal linear part of increment of the tensor components under parallel displacement corresponding to the connection. The gradient of a tensor field
+
In the theory of tensor fields on a domain of an $  n $-
 +
dimensional affine space with a connection, the gradient serves to describe the principal linear part of increment of the tensor components under parallel displacement corresponding to the connection. The gradient of a tensor field
  
<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/g/g044/g044680/g04468083.png" /></td> </tr></table>
+
$$
 +
f ( t)  = \
 +
\{ {
 +
f _ {j _ {1}  \dots j _ {q} } ^ { i _ {1} \dots i _ {p} } ( t) } : {
 +
1 \leq  i _  \alpha  , j _  \beta  \leq  n
 +
} \}
 +
$$
  
of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468084.png" /> is the tensor of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468085.png" /> with components
+
of type $  ( p, q) $
 +
is the tensor of type $  ( p, q + 1 ) $
 +
with components
  
<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/g/g044/g044680/g04468086.png" /></td> </tr></table>
+
$$
 +
\{ {
 +
\nabla _ {k} f _ {j _ {1}  \dots j _ {q} } ^ { i _ {1} \dots i _ {p} } ( t) } : {
 +
1 \leq  k, i _  \alpha  , j _  \beta  \leq  n
 +
} \}
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044680/g04468087.png" /> is the operator of absolute (covariant) differentiation (cf. [[Covariant differentiation|Covariant differentiation]]).
+
where $  \nabla _ {k} $
 +
is the operator of absolute (covariant) differentiation (cf. [[Covariant differentiation|Covariant differentiation]]).
  
 
The concept of a gradient is widely employed in many problems in mathematics, mechanics and physics. Many physical fields can be regarded as gradient fields (cf. [[Potential field|Potential field]]).
 
The concept of a gradient is widely employed in many problems in mathematics, mechanics and physics. Many physical fields can be regarded as gradient fields (cf. [[Potential field|Potential field]]).
Line 83: Line 251:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.E. Kochin,  "Vector calculus and fundamentals of tensor calculus" , Moscow  (1965)  (In Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.K. [P.K. Rashevskii] Rashewski,  "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)  {{MR|}} {{ZBL|}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.E. Kochin,  "Vector calculus and fundamentals of tensor calculus" , Moscow  (1965)  (In Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.K. [P.K. Rashevskii] Rashewski,  "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)  {{MR|}} {{ZBL|}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Fleming,  "Functions of several variables" , Addison-Wesley  (1965)  {{MR|0174675}} {{ZBL|0136.34301}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Fleming,  "Functions of several variables" , Addison-Wesley  (1965)  {{MR|0174675}} {{ZBL|0136.34301}} </TD></TR></table>

Latest revision as of 19:42, 5 June 2020


One of the fundamental concepts in vector analysis and the theory of non-linear mappings.

The gradient of a scalar function $ f $ of a vector argument $ t = ( t ^ {1} \dots t ^ {n} ) $ from a Euclidean space $ E ^ {n} $ is the derivative of $ f $ with respect to the vector argument $ t $, i.e. the $ n $- dimensional vector with components $ \partial f / \partial t ^ {i} $, $ 1 \leq i \leq n $. The following notations exist for the gradient of $ f $ at $ t _ {0} $:

$$ \mathop{\rm grad} f ( t _ {0} ),\ \ \nabla f ( t _ {0} ),\ \ \frac{\partial f ( t _ {0} ) }{\partial t } ,\ \ f ^ { \prime } ( t _ {0} ) ,\ \ \left . \frac{\partial f }{\partial t } \right | _ {t _ {0} } . $$

The gradient is a covariant vector: the components of the gradient, computed in two different coordinate systems $ t = ( t ^ {1} \dots t ^ {n} ) $ and $ \tau = ( \tau ^ {1} \dots \tau ^ {n} ) $, are connected by the relations:

$$ \frac{\partial f }{\partial t ^ {i} } ( \tau ( t)) = \ \sum _ {j = 1 } ^ { n } \frac{\partial f ( \tau ) }{\partial \tau ^ {j} } \ \frac{\partial \tau ^ {j} }{\partial t ^ {i} } . $$

The vector $ f ^ { \prime } ( t _ {0} ) $, with its origin at $ t _ {0} $, points to the direction of fastest increase of $ f $, and is orthogonal to the level lines or surfaces of $ f $ passing through $ t _ {0} $.

The derivative of the function at $ t _ {0} $ in the direction of an arbitrary unit vector $ \mathbf N = ( N ^ {1} \dots N ^ {n} ) $ is equal to the projection of the gradient function onto this direction:

$$ \tag{1 } \frac{\partial f ( t _ {0} ) }{\partial \mathbf N } = \ ( f ^ { \prime } ( t _ {0} ), \mathbf N ) \equiv \ \sum _ {j = 1 } ^ { n } \frac{\partial f ( t _ {0} ) }{\partial t ^ {j} } N ^ {j} = | f ^ { \prime } ( t _ {0} ) | \cos \phi , $$

where $ \phi $ is the angle between $ \mathbf N $ and $ f ^ { \prime } ( t _ {0} ) $. The maximal directional derivative is attained if $ \phi = 0 $, i.e. in the direction of the gradient, and that maximum is equal to the length of the gradient.

The concept of a gradient is closely connected with the concept of the differential of a function. If $ f $ is differentiable at $ t _ {0} $, then, in a neighbourhood of that point,

$$ \tag{2 } f ( t) = f ( t _ {0} ) + ( f ^ { \prime } ( t _ {0} ),\ t - t _ {0} ) + o ( | t - t _ {0} | ), $$

i.e. $ df = ( f ^ { \prime } ( t _ {0} ), dt) $. The existence of the gradient of $ f $ at $ t _ {0} $ is not sufficient for formula (2) to be valid.

A point $ t _ {0} $ at which $ f ^ { \prime } ( t _ {0} ) = 0 $ is called a stationary (critical or extremal) point of $ f $. An example of such a point is a local extremal point of $ f $, and the system $ \partial f ( t _ {0} ) / \partial t ^ {i} = 0 $, $ 1 \leq i \leq n $, is employed to find an extremal point $ t _ {0} $.

The following formulas can be used to compute the value of the gradient:

$$ \mathop{\rm grad} ( \lambda f ) = \ \lambda \mathop{\rm grad} f,\ \ \lambda = \textrm{ const } , $$

$$ \mathop{\rm grad} ( f + g) = \mathop{\rm grad} f + \mathop{\rm grad} g, $$

$$ \mathop{\rm grad} ( fg) = g \mathop{\rm grad} f + f \mathop{\rm grad} g, $$

$$ \mathop{\rm grad} \left ( { \frac{f}{g} } \right ) = \frac{1}{g ^ {2} } ( g \mathop{\rm grad} f - f \mathop{\rm grad} g). $$

The gradient $ f ^ { \prime } ( t _ {0} ) $ is the derivative at $ t _ {0} $ with respect to volume of the vector function given by

$$ \Phi ( E) = \ \int\limits _ {t \in \partial E } f ( t) \mathbf M ds, $$

where $ E $ is a domain with boundary $ \partial E $, $ t _ {0} \in E $, $ ds $ is the area element of $ \partial E $, and $ \mathbf M $ is the unit vector of the outward normal to $ \partial E $. In other words,

$$ f ^ { \prime } ( t _ {0} ) = \ \lim\limits \frac{\Phi ( E) }{ \mathop{\rm vol} E } \ \textrm{ as } \ \ E \rightarrow t _ {0} . $$

Formulas (1), (2) and the properties of the gradient listed above indicate that the concept of a gradient is invariant with respect to the choice of a coordinate system.

In a curvilinear coordinate system $ x = ( x ^ {1} \dots x ^ {n} ) $, in which the square of the linear element is

$$ ds ^ {2} = \ \sum _ {i, j = 1 } ^ { n } g _ {ij} ( x) dx ^ {i} dx ^ {j} , $$

the components of the gradient of $ f $ with respect to the unit vectors tangent to coordinate lines at $ x $ are

$$ \sum _ {j = 1 } ^ { n } g ^ {ij} ( x) \frac{\partial f }{\partial x ^ {j} } ,\ \ 1 \leq i \leq n, $$

where the matrix $ \| g ^ {ij} \| $ is the inverse of the matrix $ \| g _ {ij} \| $.

The concept of a gradient for more general vector functions of a vector argument is introduced by means of equation (2). Thus, the gradient is a linear operator the effect of which on the increment $ t - t _ {0} $ of the argument is to yield the principal linear part of the increment $ f( t) - f( t _ {0} ) $ of the vector function $ f $. E.g., if $ f = ( f ^ { 1 } \dots f ^ { m } ) $ is an $ m $- dimensional vector function of the argument $ t = ( t ^ {1} \dots t ^ {n} ) $, then its gradient at a point $ t _ {0} $ is the Jacobi matrix $ J = J ( t _ {0} ) $ with components $ ( \partial f ^ { i } / \partial t ^ {j} ) ( t _ {0} ) $, $ 1 \leq i \leq m $, $ 1 \leq j \leq n $, and

$$ f ( t) = f ( t _ {0} ) + J ( t - t _ {0} ) + o ( t - t _ {0} ), $$

where $ o ( t - t _ {0} ) $ is an $ m $- dimensional vector of length $ o ( | t - t _ {0} | ) $. The matrix $ J $ is defined by the limit transition

$$ \tag{3 } \lim\limits _ {\rho \rightarrow 0 } \ \frac{f ( t _ {0} + \rho \tau ) - f ( t _ {0} ) } \rho = J \tau , $$

for any fixed $ n $- dimensional vector $ \tau $.

In an infinite-dimensional Hilbert space definition (3) is equivalent to the definition of differentiability according to Fréchet, the gradient then being identical with the Fréchet derivative.

If the values of $ f $ lie in an infinite-dimensional vector space, various types of limit transitions in (3) are possible (see, for example, Gâteaux derivative).

In the theory of tensor fields on a domain of an $ n $- dimensional affine space with a connection, the gradient serves to describe the principal linear part of increment of the tensor components under parallel displacement corresponding to the connection. The gradient of a tensor field

$$ f ( t) = \ \{ { f _ {j _ {1} \dots j _ {q} } ^ { i _ {1} \dots i _ {p} } ( t) } : { 1 \leq i _ \alpha , j _ \beta \leq n } \} $$

of type $ ( p, q) $ is the tensor of type $ ( p, q + 1 ) $ with components

$$ \{ { \nabla _ {k} f _ {j _ {1} \dots j _ {q} } ^ { i _ {1} \dots i _ {p} } ( t) } : { 1 \leq k, i _ \alpha , j _ \beta \leq n } \} , $$

where $ \nabla _ {k} $ is the operator of absolute (covariant) differentiation (cf. Covariant differentiation).

The concept of a gradient is widely employed in many problems in mathematics, mechanics and physics. Many physical fields can be regarded as gradient fields (cf. Potential field).

References

[1] N.E. Kochin, "Vector calculus and fundamentals of tensor calculus" , Moscow (1965) (In Russian)
[2] P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)

Comments

References

[a1] W. Fleming, "Functions of several variables" , Addison-Wesley (1965) MR0174675 Zbl 0136.34301
How to Cite This Entry:
Gradient. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gradient&oldid=47110
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article