Editing Divergence theorem (section)

{{short description|Theorem in calculus which relates the flux of closed surfaces to divergence over their volume}}
{{redirect|Gauss's theorem|the theorem concerning the electric field|Gauss's law}}
{{redirect|Ostrogradsky's theorem|the theorem in mechanics|Ostrogradsky instability}}
{{Calculus |expanded=vector}}

In [[vector calculus]], the '''divergence theorem''', also known as '''Gauss's theorem''' or '''Ostrogradsky's theorem''',<ref name="Katz">{{cite journal | last = Katz | first = Victor J. | title = The history of Stokes's theorem | journal = Mathematics Magazine | volume = 52 | issue = 3| pages = 146–156 | year = 1979 | doi = 10.2307/2690275| jstor = 2690275 }} reprinted in {{cite book| last = Anderson | first = Marlow | title = Who Gave You the Epsilon?: And Other Tales of Mathematical History | publisher = Mathematical Association of America | year = 2009 | pages = 78–79 | url = https://books.google.com/books?id=WwFMjsym9JwC&q=%22ostrogradsky's+theorem&pg=PA78 | isbn = 978-0-88385-569-0}}</ref> is a [[theorem]] relating the ''[[flux]]'' of a [[vector field]] through a closed [[surface (mathematics)|surface]] to the ''[[divergence]]'' of the field in the volume enclosed.

More precisely, the divergence theorem states that the [[surface integral]] of a vector field over a closed surface, which is called the "flux" through the surface, is equal to the [[volume integral]] of the divergence over the region enclosed by the surface. Intuitively, it states that "the sum of all sources of the field in a region (with sinks regarded as negative sources) gives the net flux out of the region".

The divergence theorem is an important result for the mathematics of [[physics]] and [[engineering]], particularly in [[electrostatics]] and [[fluid dynamics]]. In these fields, it is usually applied in three dimensions. However, it [[Generalization|generalizes]] to any number of dimensions. In one dimension, it is equivalent to the [[fundamental theorem of calculus]]. In two dimensions, it is equivalent to [[Green's theorem]].