Flux of a vector field

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A concept in the theory of vector fields. The flux of a vector field through the surface is expressed, up to sign, by the surface integral

where is the unit normal vector to the surface (it is assumed that the vector changes continuously over the surface ). The flux of the velocity field of a fluid is equal to the volume of fluid passing through the surface per unit time.


The flux of a differential vector field (defined by the formula above) is related to the divergence of :

where is the volume element in and is the Hamilton operator, . This equation is called the divergence theorem or also Green's theorem in space, cf. [a1] and Stokes theorem.


[a1] M. Spivak, "Calculus on manifolds" , Benjamin (1965)
How to Cite This Entry:
Flux of a vector field. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article