Vector field, source of a

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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 $.


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


[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:,_source_of_a&oldid=49137
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article