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Divergence theorem

The volume integral of the divergence of a vector field over a region equals the total outward flux of the field through the closed surface bounding that region. It is used to convert between volume and surface integrals in vector calculus and electromagnetism.

\iiint_V (\nabla \cdot \boldsymbol{F}) \, \mathrm{d}V = \iint_{\partial V} \boldsymbol{F} \cdot \mathrm{d}\boldsymbol{\sigma}

What the symbols mean

volume region in space bounded by the closed surface
closed boundary surface of V
vector field defined on V
divergence of the vector field F
volume element
outward-pointing area element vector on the boundary surface

About this formula

  • FieldMathematics
  • TopicVector calculus
  • LevelUniversity
  • Also calledGauss's theorem, Ostrogradsky's theorem, Gauss divergence theorem
  • SourceWikidata Q338886

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