Stokes' Theorem on Manifolds

Stokes’ theorem reveals a profound duality: derivatives and boundaries are opposites, not derivatives and integrals. This insight unifies calculus by showing that removing derivatives equals taking boundaries, and vice versa.

The exterior derivative d is the universal differentiation operator on manifolds, generalizing gradients, curls, and divergences from vector calculus. It acts on differential forms, providing the “little changes” measured in Stokes’ theorem.

The generalized Stokes’ theorem unifies all of calculus into one statement: ∫M dω = ∫{∂M} ω. This elegant formula subsumes the fundamental theorem of calculus, Green’s theorem, and the divergence theorem as special cases.