Stokes’s 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 R3. 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.
Stokes’ theorem is a special case of the generalized Stokes’ theorem.[5][6] In particular, a vector field on R3 can be considered as a 1-form in which case its curl is its exterior derivative, a 2-form.
Contents
1 Theorem
2 Proof
2.1 Elementary proof
2.1.1 First step of the elementary proof (parametrization of integral)
2.1.2 Second step in the elementary proof (defining the pullback)
2.1.3 Third step of the elementary proof (second equation)
2.1.4 Fourth step of the elementary proof (reduction to Green’s theorem)
2.2 Proof via differential forms
3 Applications
3.1 Irrotational fields
3.1.1 The Helmholtz’s theorems
3.1.1.1 Proof of the Helmholtz’s theorems
3.2 Conservative forces
3.3 Maxwell’s equations