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Divergence theorem and Stokes

The divergence theorem (Gauss) relates the flux out of a closed surface to the volume integral of the divergence inside. Stokes' theorem is the analogue for an open surface: the circulation along the boundary curve equals the flux of the curl through the surface.

∯SF⃗⋅dS⃗=∭V∇⋅F⃗ dV\oiint_S \vec F\cdot d\vec S = \iiint_V \nabla\cdot\vec F\,dVthe divergence theorem (Gauss)
∮CF⃗⋅dr⃗=∬S(∇×F⃗)⋅dS⃗\oint_C \vec F\cdot d\vec r = \iint_S(\nabla\times\vec F)\cdot d\vec SStokes' theorem

Symbols

SSsurface (closed for Gauss, with boundary C for Stokes)
VVthe volume inside S

Example

∇⋅F⃗=2\nabla\cdot\vec F=2 everywhere: the flux out of a cube with side 2 is 2⋅23=162\cdot2^3=16.

Green's theorem in the plane is the special case of Stokes' theorem.
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Part of Multivariable Calculus: Vector calculus.