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. It is illustrated in the figure, where the direction of positive circulation of the bounding contour ∂Σ, and the direction n of positive flux through the surface Σ, are related by a right-hand-rule. For the right hand the fingers circulate along ∂Σ and the thumb is directed along n.

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.