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A branch of [[Vector calculus|vector calculus]] in which scalar and vector fields are studied (cf. [[Scalar field|Scalar field]]; [[Vector field|Vector field]]).
 
A branch of [[Vector calculus|vector calculus]] in which scalar and vector fields are studied (cf. [[Scalar field|Scalar field]]; [[Vector field|Vector field]]).
  
One of the fundamental concepts in vector analysis for the study of scalar fields is the [[Gradient|gradient]]. A scalar field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v0963601.png" /> is said to be differentiable at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v0963602.png" /> of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v0963603.png" /> if the increment of the field, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v0963604.png" />, at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v0963605.png" /> may be written as
+
One of the fundamental concepts in vector analysis for the study of scalar fields is the [[Gradient|gradient]]. A scalar field $  u( M) $
 +
is said to be differentiable at a point $  M $
 +
of a domain $  D $
 +
if the increment of the field, $  \Delta u $,  
 +
at $  M $
 +
may be written as
  
<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/v/v096/v096360/v0963606.png" /></td> </tr></table>
+
$$
 +
\Delta u  = f ( \Delta {\mathbf r } ) + o ( \rho ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v0963607.png" /> is the vector connecting the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v0963608.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v0963609.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636010.png" /> is the distance between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636013.png" /> is a linear form applied to the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636014.png" />. The linear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636015.png" /> may be uniquely represented as
+
where $  \mathbf r $
 +
is the vector connecting the points $  M $
 +
and $  M  ^  \prime  $,
 +
$  \rho = \rho ( M, M  ^  \prime  ) $
 +
is the distance between $  M $
 +
and $  M  ^  \prime  $
 +
and $  f( \Delta \mathbf r ) $
 +
is a linear form applied to the vector $  \Delta \mathbf r $.  
 +
The linear form $  f( \Delta \mathbf r ) $
 +
may be uniquely represented as
  
<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/v/v096/v096360/v09636016.png" /></td> </tr></table>
+
$$
 +
f ( \Delta \mathbf r )  = ( \mathbf g , \Delta \mathbf r ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636017.png" /> is a vector which does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636018.png" /> (i.e. on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636019.png" />). The vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636020.png" /> is said to be the gradient of the scalar field and is denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636021.png" />. If the scalar field is differentiable at every point of some domain, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636022.png" /> is a vector field. The direction of the gradient is always orthogonal to the level lines (surfaces) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636023.png" /> of the scalar field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636024.png" />, with the directional derivative given by
+
where $  \mathbf g $
 +
is a vector which does not depend on $  \Delta \mathbf r $(
 +
i.e. on the choice of $  M  ^  \prime  $).  
 +
The vector $  \mathbf g $
 +
is said to be the gradient of the scalar field and is denoted by the symbol $  \mathop{\rm grad}  u $.  
 +
If the scalar field is differentiable at every point of some domain, $  \mathop{\rm grad}  u $
 +
is a vector field. The direction of the gradient is always orthogonal to the level lines (surfaces) $  u( M) = \textrm{ const } $
 +
of the scalar field $  u $,  
 +
with the directional derivative given by
  
<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/v/v096/v096360/v09636025.png" /></td> </tr></table>
+
$$
 +
\nabla _ {e} u  = (  \mathop{\rm grad}  u , \mathbf e ).
 +
$$
  
The concepts of [[Divergence|divergence]] and [[Curl|curl]] are also employed in the study of vector fields. Let a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636026.png" /> be differentiable at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636027.png" /> of a certain domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636028.png" />, i.e. the field increment at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636029.png" /> can be uniquely represented as
+
The concepts of [[Divergence|divergence]] and [[Curl|curl]] are also employed in the study of vector fields. Let a vector field $  \mathbf a ( M) $
 +
be differentiable at a point $  M $
 +
of a certain domain $  D $,  
 +
i.e. the field increment at the point $  M $
 +
can be uniquely represented as
  
<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/v/v096/v096360/v09636030.png" /></td> </tr></table>
+
$$
 +
\Delta {\mathbf a }  = A \Delta {\mathbf r } + o ( | \Delta {\mathbf r } | ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636032.png" /> is a linear operator which is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636033.png" /> (of the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636034.png" />). The divergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636035.png" /> of the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636036.png" /> is the following scalar invariant of the linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636037.png" />:
+
where $  \Delta {\mathbf r } = | M M  ^  \prime  | $
 +
and $  A $
 +
is a linear operator which is independent of $  \Delta \mathbf r $(
 +
of the choice of $  M  ^  \prime  $).  
 +
The divergence $  \mathop{\rm div}  \mathbf a $
 +
of the vector field $  \mathbf a ( M) $
 +
is the following scalar invariant of the linear operator $  A $:
  
<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/v/v096/v096360/v09636038.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
\mathop{\rm div}  \mathbf a  \equiv  ( \mathbf r  ^ {i} , A \mathbf r _ {i} ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636039.png" /> are dual bases: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636040.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636041.png" /> is the Kronecker symbol). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636042.png" /> is the velocity field of a stationary flow of a non-compressible liquid, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636043.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636044.png" /> denotes the intensity of the source (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636045.png" />) or of the sink (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636046.png" />) present at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636047.png" />, or their absence (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636048.png" />).
+
where $  \mathbf r  ^ {i} , \mathbf r _ {i} $
 +
are dual bases: $  ( \mathbf r _ {i} , \mathbf r  ^ {k} ) = \delta _ {i}  ^ {k} $(
 +
$  \delta _ {i}  ^ {k} $
 +
is the Kronecker symbol). If $  \mathbf a ( M) $
 +
is the velocity field of a stationary flow of a non-compressible liquid, $  \mathop{\rm div}  \mathbf a $
 +
at the point $  M $
 +
denotes the intensity of the source ( $  \mathop{\rm div}  \mathbf a > 0 $)  
 +
or of the sink ( $  \mathop{\rm div}  \mathbf a < 0 $)  
 +
present at $  M $,  
 +
or their absence ( $  \mathop{\rm div}  \mathbf a = 0 $).
  
The curl (rotor) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636049.png" /> of the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636050.png" /> on a domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636051.png" /> is the following vector invariant of the linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636052.png" /> from (*):
+
The curl (rotor) $  \mathop{\rm rot}  \mathbf a $
 +
of the vector field $  \mathbf a ( M) $
 +
on a domain in $  \mathbf R  ^ {3} $
 +
is the following vector invariant of the linear operator $  A $
 +
from (*):
  
<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/v/v096/v096360/v09636053.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm rot}  \mathbf a  \equiv  [ \mathbf r _ {i} , A \mathbf r  ^ {i} ],
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636054.png" /> are dual bases. The curl of a vector field may be interpreted as the  "rotational component"  of this field.
+
where $  \mathbf r  ^ {i} , \mathbf r _ {i} $
 +
are dual bases. The curl of a vector field may be interpreted as the  "rotational component"  of this field.
  
For vector and scalar fields of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636055.png" /> repeated operations are possible, for example:
+
For vector and scalar fields of class $  C  ^ {2} $
 +
repeated operations are possible, for example:
  
<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/v/v096/v096360/v09636056.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm rot}  \mathop{\rm grad}  u  = 0,
 +
$$
  
<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/v/v096/v096360/v09636057.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm div}  \mathop{\rm rot}  \mathbf a  = 0,
 +
$$
  
<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/v/v096/v096360/v09636058.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm rot}  \mathop{\rm rot}  \mathbf a  =   \mathop{\rm grad}  \mathop{\rm div}  \mathbf a - \Delta {\mathbf a } ,
 +
$$
  
<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/v/v096/v096360/v09636059.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm div}  \mathop{\rm grad}  u  = \Delta  u ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636060.png" /> is the [[Laplace operator|Laplace operator]].
+
where $  \Delta $
 +
is the [[Laplace operator|Laplace operator]].
  
 
Gradient, divergence and curl together are usually known as the basic differential operations of vector analysis. See [[Curl|Curl]]; [[Gradient|Gradient]]; [[Divergence|Divergence]] for their properties and expressions in special coordinate systems.
 
Gradient, divergence and curl together are usually known as the basic differential operations of vector analysis. See [[Curl|Curl]]; [[Gradient|Gradient]]; [[Divergence|Divergence]] for their properties and expressions in special coordinate systems.
  
Fundamental integral formulas, connecting volume, surface and contour integrals, can be written down in terms of the basic operations of vector analysis. Let a vector field be continuously differentiable in a bounded connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636061.png" /> with piecewise-smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636062.png" />.
+
Fundamental integral formulas, connecting volume, surface and contour integrals, can be written down in terms of the basic operations of vector analysis. Let a vector field be continuously differentiable in a bounded connected domain $  V $
 
+
with piecewise-smooth boundary $  L $.
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636063.png" /> be a bounded, complete, piecewise-smooth, two-sided (oriented) surface with piecewise-smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636064.png" />. Then the [[Stokes formula|Stokes formula]] will be applicable:
 
 
 
<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/v/v096/v096360/v09636065.png" /></td> </tr></table>
 
  
where the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636066.png" /> normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636067.png" /> and the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636068.png" /> tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636069.png" /> must be determined in accordance with the orientations of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636070.png" /> and its boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636071.png" />. The integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636072.png" /> is known as the circulation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636073.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636074.png" />. If the circulation of a vector field along an arbitrary closed piecewise-smooth curve in a given domain is zero, the vector field is said to be potential (or conservative) in this domain. In a simply-connected domain a vector field is conservative if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636075.png" />. For a conservative vector field there exists the so-called scalar potential, which is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636076.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636077.png" />; here
+
Let  $  S $
 +
be a bounded, complete, piecewise-smooth, two-sided (oriented) surface with piecewise-smooth boundary  $  \partial  S $.  
 +
Then the [[Stokes formula|Stokes formula]] will be applicable:
  
<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/v/v096/v096360/v09636078.png" /></td> </tr></table>
+
$$
 +
{\int\limits \int\limits } _ { S } ( \mathbf n ,  \mathop{\rm rot}  {\mathbf a } )  ds  = \
 +
\oint _  \partial  S ( \mathbf a , \mathbf t )  dl,
 +
$$
  
where the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636080.png" /> is a piecewise-smooth curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636082.png" /> is the unit vector tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636083.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636084.png" /> is the line element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636085.png" />.
+
where the vector  $  \mathbf n $
 +
normal to  $  S $
 +
and the vector  $  \mathbf t $
 +
tangent to  $  \partial  S $
 +
must be determined in accordance with the orientations of the surface  $  S $
 +
and its boundary  $  \partial  S $.  
 +
The integral  $  \oint _ {\partial  S }  ( \mathbf a , \mathbf t )  dl $
 +
is known as the circulation of  $  \mathbf a $
 +
along  $  \partial  S $.  
 +
If the circulation of a vector field along an arbitrary closed piecewise-smooth curve in a given domain is zero, the vector field is said to be potential (or conservative) in this domain. In a simply-connected domain a vector field is conservative if  $  \mathop{\rm rot}  \mathbf a = 0 $.  
 +
For a conservative vector field there exists the so-called scalar potential, which is a function  $  v ( M) $
 +
such that  $  \mathbf a = \mathop{\rm grad}  v $;
 +
here
  
Let the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636086.png" /> be continuously differentiable in a bounded connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636087.png" /> with piecewise-smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636088.png" />; the [[Ostrogradski formula|Ostrogradski formula]] reads as follows:
+
$$
 +
\int\limits _ { AB } ( \mathbf a , \mathbf t )  dl  = v( B) - v( A),
 +
$$
  
<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/v/v096/v096360/v09636089.png" /></td> </tr></table>
+
where the points  $  A, B \in D $,
 +
$  AB $
 +
is a piecewise-smooth curve in  $  D $,
 +
$  \mathbf t $
 +
is the unit vector tangent to  $  AB $,
 +
and  $  dl $
 +
is the line element of  $  AB $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636090.png" /> is the exterior normal vector to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636091.png" />.
+
Let the vector field  $  \mathbf a ( M) $
 +
be continuously differentiable in a bounded connected domain  $  V $
 +
with piecewise-smooth boundary  $  \partial  V $;
 +
the [[Ostrogradski formula|Ostrogradski formula]] reads as follows:
  
The integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636092.png" /> is said to be the flux of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636093.png" /> across <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636094.png" />. If the flux of a vector field across an arbitrary, piecewise-smooth, non-self-intersecting, oriented surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636095.png" /> which is the boundary of some bounded subdomain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636096.png" /> is zero, the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636097.png" /> is said to be solenoidal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636098.png" />. For a continuously-differentiable vector field to be solenoidal it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v09636099.png" /> at all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v096360100.png" />. For a solenoidal vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v096360101.png" /> there exists a so-called vector potential: a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v096360102.png" /> such that
+
$$
 +
{\int\limits \int\limits \int\limits } _ { V }  \mathop{\rm div}  \mathbf a d \sigma  = \
 +
{\int\limits \int\limits } _  \partial  V ( \mathbf n , \mathbf a )  ds,
 +
$$
  
<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/v/v096/v096360/v096360103.png" /></td> </tr></table>
+
where  $  \mathbf n $
 +
is the exterior normal vector to  $  \partial  V $.
  
If the divergence and the curl of a vector field are defined at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v096360104.png" /> of a simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v096360105.png" />, the vector field can be represented everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v096360106.png" /> as the sum of a potential field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v096360107.png" /> and a solenoidal field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v096360108.png" /> (Helmholtz' theorem):
+
The integral  $  \int \int _ {\partial  V }  ( \mathbf n , \mathbf a )  ds $
 +
is said to be the flux of  $  \mathbf a ( M) $
 +
across  $  \partial  V $.
 +
If the flux of a vector field across an arbitrary, piecewise-smooth, non-self-intersecting, oriented surface in  $  V $
 +
which is the boundary of some bounded subdomain of $  V $
 +
is zero, the vector field $  \mathbf a ( M) $
 +
is said to be solenoidal in $  V $.  
 +
For a continuously-differentiable vector field to be solenoidal it is necessary and sufficient that  $  \mathop{\rm div}  \mathbf a = 0 $
 +
at all points of  $  V $.  
 +
For a solenoidal vector field $  \mathbf a ( M) $
 +
there exists a so-called vector potential: a function  $  A( M) $
 +
such that
  
<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/v/v096/v096360/v096360109.png" /></td> </tr></table>
+
$$
 +
\mathbf a  =   \mathop{\rm rot}  A( M).
 +
$$
  
Vector fields for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v096360110.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v096360111.png" /> are called harmonic. The potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v096360112.png" /> of a harmonic vector field satisfies the Laplace equation. The scalar field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v096360113.png" /> is also said to be harmonic. For references, see [[Vector calculus|Vector calculus]].
+
If the divergence and the curl of a vector field are defined at each point  $  M $
 +
of a simply-connected domain  $  D $,
 +
the vector field can be represented everywhere in  $  D $
 +
as the sum of a potential field $  \mathbf a _ {1} ( M) $
 +
and a solenoidal field  $  \mathbf a _ {2} ( M) $(
 +
Helmholtz' theorem):
  
 +
$$
 +
\mathbf a ( M)  =  \mathbf a _ {1} ( M)+ \mathbf a _ {2} ( M).
 +
$$
  
 +
Vector fields for which  $  \mathop{\rm div}  \mathbf a = 0 $
 +
and  $  \mathop{\rm rot}  \mathbf a = 0 $
 +
are called harmonic. The potential  $  v $
 +
of a harmonic vector field satisfies the Laplace equation. The scalar field  $  v $
 +
is also said to be harmonic. For references, see [[Vector calculus|Vector calculus]].
  
 
====Comments====
 
====Comments====
 
Ostrogradski's formula is commonly called Gauss' formula.
 
Ostrogradski's formula is commonly called Gauss' formula.
  
The condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v096360114.png" /> is necessary for a vector field to be solenoidal. It is sufficient on, for example, convex domains. The general additional condition is that the second homology of the domain vanishes. This can easily be seen from the de Rham cohomology theory. There are examples of vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v096360115.png" />-space with one point removed which have vanishing divergence, but are not solenoidal.
+
The condition $  \mathop{\rm div}  a = 0 $
 +
is necessary for a vector field to be solenoidal. It is sufficient on, for example, convex domains. The general additional condition is that the second homology of the domain vanishes. This can easily be seen from the de Rham cohomology theory. There are examples of vector fields on $  3 $-
 +
space with one point removed which have vanishing divergence, but are not solenoidal.
  
The notions of gradient, divergence, Laplace operator, flux of a vector field, and the given integral formulas can easily be extended to higher-dimensional Euclidean spaces and Riemannian manifolds, and all other notions can be extended to Riemannian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v096360116.png" />-manifolds.
+
The notions of gradient, divergence, Laplace operator, flux of a vector field, and the given integral formulas can easily be extended to higher-dimensional Euclidean spaces and Riemannian manifolds, and all other notions can be extended to Riemannian $  3 $-
 +
manifolds.
  
In this context, the given integral formulas appear in a unified way as Stokes' formula, saying that the integral of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v096360117.png" />-form over the piecewise-regular boundary of a smooth orientable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096360/v096360118.png" />-submanifold is equal to the integral of its exterior differential over the submanifold itself.
+
In this context, the given integral formulas appear in a unified way as Stokes' formula, saying that the integral of a $  k $-
 +
form over the piecewise-regular boundary of a smooth orientable $  ( k- 1) $-
 +
submanifold is equal to the integral of its exterior differential over the submanifold itself.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Marsden,  "Calculus" , '''3''' , Springer  (1988)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.J. Hicks,  "Notes on differential geometry" , v. Nostrand  (1965)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Marsden,  "Calculus" , '''3''' , Springer  (1988)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.J. Hicks,  "Notes on differential geometry" , v. Nostrand  (1965)</TD></TR></table>

Latest revision as of 08:28, 6 June 2020


A branch of vector calculus in which scalar and vector fields are studied (cf. Scalar field; Vector field).

One of the fundamental concepts in vector analysis for the study of scalar fields is the gradient. A scalar field $ u( M) $ is said to be differentiable at a point $ M $ of a domain $ D $ if the increment of the field, $ \Delta u $, at $ M $ may be written as

$$ \Delta u = f ( \Delta {\mathbf r } ) + o ( \rho ) , $$

where $ \mathbf r $ is the vector connecting the points $ M $ and $ M ^ \prime $, $ \rho = \rho ( M, M ^ \prime ) $ is the distance between $ M $ and $ M ^ \prime $ and $ f( \Delta \mathbf r ) $ is a linear form applied to the vector $ \Delta \mathbf r $. The linear form $ f( \Delta \mathbf r ) $ may be uniquely represented as

$$ f ( \Delta \mathbf r ) = ( \mathbf g , \Delta \mathbf r ), $$

where $ \mathbf g $ is a vector which does not depend on $ \Delta \mathbf r $( i.e. on the choice of $ M ^ \prime $). The vector $ \mathbf g $ is said to be the gradient of the scalar field and is denoted by the symbol $ \mathop{\rm grad} u $. If the scalar field is differentiable at every point of some domain, $ \mathop{\rm grad} u $ is a vector field. The direction of the gradient is always orthogonal to the level lines (surfaces) $ u( M) = \textrm{ const } $ of the scalar field $ u $, with the directional derivative given by

$$ \nabla _ {e} u = ( \mathop{\rm grad} u , \mathbf e ). $$

The concepts of divergence and curl are also employed in the study of vector fields. Let a vector field $ \mathbf a ( M) $ be differentiable at a point $ M $ of a certain domain $ D $, i.e. the field increment at the point $ M $ can be uniquely represented as

$$ \Delta {\mathbf a } = A \Delta {\mathbf r } + o ( | \Delta {\mathbf r } | ), $$

where $ \Delta {\mathbf r } = | M M ^ \prime | $ and $ A $ is a linear operator which is independent of $ \Delta \mathbf r $( of the choice of $ M ^ \prime $). The divergence $ \mathop{\rm div} \mathbf a $ of the vector field $ \mathbf a ( M) $ is the following scalar invariant of the linear operator $ A $:

$$ \tag{* } \mathop{\rm div} \mathbf a \equiv ( \mathbf r ^ {i} , A \mathbf r _ {i} ), $$

where $ \mathbf r ^ {i} , \mathbf r _ {i} $ are dual bases: $ ( \mathbf r _ {i} , \mathbf r ^ {k} ) = \delta _ {i} ^ {k} $( $ \delta _ {i} ^ {k} $ is the Kronecker symbol). If $ \mathbf a ( M) $ is the velocity field of a stationary flow of a non-compressible liquid, $ \mathop{\rm div} \mathbf a $ at the point $ M $ denotes the intensity of the source ( $ \mathop{\rm div} \mathbf a > 0 $) or of the sink ( $ \mathop{\rm div} \mathbf a < 0 $) present at $ M $, or their absence ( $ \mathop{\rm div} \mathbf a = 0 $).

The curl (rotor) $ \mathop{\rm rot} \mathbf a $ of the vector field $ \mathbf a ( M) $ on a domain in $ \mathbf R ^ {3} $ is the following vector invariant of the linear operator $ A $ from (*):

$$ \mathop{\rm rot} \mathbf a \equiv [ \mathbf r _ {i} , A \mathbf r ^ {i} ], $$

where $ \mathbf r ^ {i} , \mathbf r _ {i} $ are dual bases. The curl of a vector field may be interpreted as the "rotational component" of this field.

For vector and scalar fields of class $ C ^ {2} $ repeated operations are possible, for example:

$$ \mathop{\rm rot} \mathop{\rm grad} u = 0, $$

$$ \mathop{\rm div} \mathop{\rm rot} \mathbf a = 0, $$

$$ \mathop{\rm rot} \mathop{\rm rot} \mathbf a = \mathop{\rm grad} \mathop{\rm div} \mathbf a - \Delta {\mathbf a } , $$

$$ \mathop{\rm div} \mathop{\rm grad} u = \Delta u , $$

where $ \Delta $ is the Laplace operator.

Gradient, divergence and curl together are usually known as the basic differential operations of vector analysis. See Curl; Gradient; Divergence for their properties and expressions in special coordinate systems.

Fundamental integral formulas, connecting volume, surface and contour integrals, can be written down in terms of the basic operations of vector analysis. Let a vector field be continuously differentiable in a bounded connected domain $ V $ with piecewise-smooth boundary $ L $.

Let $ S $ be a bounded, complete, piecewise-smooth, two-sided (oriented) surface with piecewise-smooth boundary $ \partial S $. Then the Stokes formula will be applicable:

$$ {\int\limits \int\limits } _ { S } ( \mathbf n , \mathop{\rm rot} {\mathbf a } ) ds = \ \oint _ \partial S ( \mathbf a , \mathbf t ) dl, $$

where the vector $ \mathbf n $ normal to $ S $ and the vector $ \mathbf t $ tangent to $ \partial S $ must be determined in accordance with the orientations of the surface $ S $ and its boundary $ \partial S $. The integral $ \oint _ {\partial S } ( \mathbf a , \mathbf t ) dl $ is known as the circulation of $ \mathbf a $ along $ \partial S $. If the circulation of a vector field along an arbitrary closed piecewise-smooth curve in a given domain is zero, the vector field is said to be potential (or conservative) in this domain. In a simply-connected domain a vector field is conservative if $ \mathop{\rm rot} \mathbf a = 0 $. For a conservative vector field there exists the so-called scalar potential, which is a function $ v ( M) $ such that $ \mathbf a = \mathop{\rm grad} v $; here

$$ \int\limits _ { AB } ( \mathbf a , \mathbf t ) dl = v( B) - v( A), $$

where the points $ A, B \in D $, $ AB $ is a piecewise-smooth curve in $ D $, $ \mathbf t $ is the unit vector tangent to $ AB $, and $ dl $ is the line element of $ AB $.

Let the vector field $ \mathbf a ( M) $ be continuously differentiable in a bounded connected domain $ V $ with piecewise-smooth boundary $ \partial V $; the Ostrogradski formula reads as follows:

$$ {\int\limits \int\limits \int\limits } _ { V } \mathop{\rm div} \mathbf a d \sigma = \ {\int\limits \int\limits } _ \partial V ( \mathbf n , \mathbf a ) ds, $$

where $ \mathbf n $ is the exterior normal vector to $ \partial V $.

The integral $ \int \int _ {\partial V } ( \mathbf n , \mathbf a ) ds $ is said to be the flux of $ \mathbf a ( M) $ across $ \partial V $. If the flux of a vector field across an arbitrary, piecewise-smooth, non-self-intersecting, oriented surface in $ V $ which is the boundary of some bounded subdomain of $ V $ is zero, the vector field $ \mathbf a ( M) $ is said to be solenoidal in $ V $. For a continuously-differentiable vector field to be solenoidal it is necessary and sufficient that $ \mathop{\rm div} \mathbf a = 0 $ at all points of $ V $. For a solenoidal vector field $ \mathbf a ( M) $ there exists a so-called vector potential: a function $ A( M) $ such that

$$ \mathbf a = \mathop{\rm rot} A( M). $$

If the divergence and the curl of a vector field are defined at each point $ M $ of a simply-connected domain $ D $, the vector field can be represented everywhere in $ D $ as the sum of a potential field $ \mathbf a _ {1} ( M) $ and a solenoidal field $ \mathbf a _ {2} ( M) $( Helmholtz' theorem):

$$ \mathbf a ( M) = \mathbf a _ {1} ( M)+ \mathbf a _ {2} ( M). $$

Vector fields for which $ \mathop{\rm div} \mathbf a = 0 $ and $ \mathop{\rm rot} \mathbf a = 0 $ are called harmonic. The potential $ v $ of a harmonic vector field satisfies the Laplace equation. The scalar field $ v $ is also said to be harmonic. For references, see Vector calculus.

Comments

Ostrogradski's formula is commonly called Gauss' formula.

The condition $ \mathop{\rm div} a = 0 $ is necessary for a vector field to be solenoidal. It is sufficient on, for example, convex domains. The general additional condition is that the second homology of the domain vanishes. This can easily be seen from the de Rham cohomology theory. There are examples of vector fields on $ 3 $- space with one point removed which have vanishing divergence, but are not solenoidal.

The notions of gradient, divergence, Laplace operator, flux of a vector field, and the given integral formulas can easily be extended to higher-dimensional Euclidean spaces and Riemannian manifolds, and all other notions can be extended to Riemannian $ 3 $- manifolds.

In this context, the given integral formulas appear in a unified way as Stokes' formula, saying that the integral of a $ k $- form over the piecewise-regular boundary of a smooth orientable $ ( k- 1) $- submanifold is equal to the integral of its exterior differential over the submanifold itself.

References

[a1] A. Marsden, "Calculus" , 3 , Springer (1988)
[a2] N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965)
How to Cite This Entry:
Vector analysis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vector_analysis&oldid=49133
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article