Stokes' theorem

Theorem in vector calculus

Stokes' theorem, also known as the Kelvin–Stokes theorem, is a theorem in vector calculus that relates the behavior of a vector field along the edge of a surface to the behavior of its curl on the surface itself. In its usual three-dimensional form, it says that the total circulation of a vector field around a closed curve is equal to the total curl of the field through a surface bounded by that curve. If Σ is an oriented surface with boundary ∂Σ, Stokes' theorem is commonly written as ∮ ∂ Σ F ⋅ d r = ∬ Σ ( ∇ × F) ⋅ n d S .

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