View source for Vector field, source of a

<!--
v0964401.png
$#A+1 = 13 n = 0
$#C+1 = 13 : ~/encyclopedia/old_files/data/V096/V.0906440 Vector field, source of a
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}}

A point of the [[Vector field|vector field]]  $  \mathbf a $
with the property that the flow of the field through any sufficiently small closed surface  $  \partial  V $
enclosing it is independent of the surface and positive. The flow

$$ 
Q  =  {\int\limits \int\limits } _ {\partial  V } ( \mathbf n , \mathbf a )  d s ,
$$

where  $  \mathbf n $
is the outward unit normal to  $  \partial  V $
and  $  s $
is the area element of  $  \partial  V $, 
is called the power of the source. If  $  Q $
is negative, one speaks of a sink. If the sources are continuously distributed over the domain  $  V $
considered, then the limit

$$ 
\lim\limits _ {\partial  V \rightarrow M } \ 

\frac{\int\limits \int\limits _ {\partial  V }  ( \mathbf a , \mathbf n )  d s }{V}

$$

is called the density (intensity) of the source at the point  $  M $. 
It is equal to the [[Divergence|divergence]] of  $  \mathbf a $
at  $  M $.

====Comments====
A combination of a source and a vortex in a hydrodynamical flow gives rise to a swirl flow.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Marsden,   A. Weinstein,   "Calculus" , '''3''' , Springer  (1988)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Triebel,   "Analysis and mathematical physics" , Reidel  (1986)  pp. Sect. 16</TD></TR></table>