PlanetPhysics/Stokes Theorem

{\mathbf Stokes' theorem}:\, The surface integral of the curl of a vector function is equal to the line integral of that vector function taken around the closed curve bounding that surface.

$$ \int \int_S \nabla \times {\mathbf W} \cdot d {\mathbf a} = \oint_O {\mathbf W} \cdot d {\mathbf r} $$

On account of its great importance in all branches of mathematical physics, a number of different proofs of this theorem will be given.

Proof I.
soon to come...