Difference between revisions of "Vector field, source of a"
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| − | + | 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 | ||
| − | + | $$ | |
| + | Q = {\int\limits \int\limits } _ {\partial V } ( \mathbf n , \mathbf a ) d s , | ||
| + | $$ | ||
| − | is called the | + | 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==== | ====Comments==== | ||
Latest revision as of 08:28, 6 June 2020
A point of the 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 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
| [a1] | J. Marsden, A. Weinstein, "Calculus" , 3 , Springer (1988) |
| [a2] | H. Triebel, "Analysis and mathematical physics" , Reidel (1986) pp. Sect. 16 |
How to Cite This Entry:
Vector field, source of a. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vector_field,_source_of_a&oldid=15332
Vector field, source of a. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vector_field,_source_of_a&oldid=15332
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article