The flux of a vector field out through a closed surface equals the integral of the field's divergence over the volume the surface encloses. Also called Gauss's theorem, it is a special case of the general Stokes' theorem central to vector calculus and physics.
Facts
StatementThe divergence theorem states that the outward flux of a vector field through a closed surface, that is, the surface integral of the field over the surface, equals the volume integral of the field's divergence over the region the surface encloses. 1 Classification
Statement Form Connections
Sources
1. Divergence Theorem (Wikipedia)
Wikimedia FoundationLead section, opening sentence
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.
History section, on Ostrogradsky's general proof
Mikhail Ostrogradsky, who gave the first proof of the general theorem, in 1826, as part of his investigation of heat flow.
Lead section, statement-form reference
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.
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