# Vector field, source of a

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=49137