Stokes' theorem

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Stokes' theorem,^[1] also known as the Kelvin–Stokes theorem^[2][3] after Lord Kelvin and George Stokes, the fundamental theorem for curls or simply the curl theorem,^[4] is a theorem in vector calculus on ${\displaystyle \mathbb {R} ^{3}}$. Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. The classical theorem of Stokes can be stated in one sentence:

The line integral of a vector field over a loop is equal to the surface integral of its curl over the enclosed surface.

Stokes' theorem is a special case of the generalized Stokes theorem.^[5][6] In particular, a vector field on ${\displaystyle \mathbb {R} ^{3}}$ can be considered as a 1-form in which case its curl is its exterior derivative, a 2-form.