# Stokes' theorem

**Stokes' theorem**,[1] also known as **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 . 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 Stokes' theorem can be stated in one sentence: The line integral of a vector field over a loop is equal to the *flux of its curl* through the enclosed surface.

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Stokes' theorem is a special case of the generalized Stokes' theorem.[5][6] In particular, a vector field on can be considered as a 1-form in which case its curl is its exterior derivative, a 2-form.