the generalized Stokes' theorem
/ STOHKS /
Stokes' theorem is the one law that unifies almost every integral theorem of calculus into a single sentence: the integral of a derivative over a region equals the integral of the original over the boundary of that region. The fundamental theorem of calculus, Green's theorem, the divergence (Gauss) theorem, and the classical Stokes (curl) theorem are all the very same statement, read in different dimensions and degrees. It is the reason differential forms were invented.
Precisely: let M be an oriented smooth n-manifold with boundary partial M (given the induced orientation), and let omega be a smooth, compactly supported (n-1)-form on M. Then the integral over M of d omega equals the integral over partial M of omega (the restriction/pullback of omega to the boundary). In symbols, integral over M of d omega = integral over partial M of omega. The exterior derivative on the left becomes a boundary operation on the right — differentiation inside is paired with taking boundary outside. When partial M is empty (M is closed), the right side is zero, so the integral of any exact top form over a closed manifold vanishes — a fact with deep cohomological consequences.
Its power is unification and abstraction. Take n = 1, M = [a, b]: it is the fundamental theorem of calculus, integral of f' equals f(b) - f(a). Take n = 2 in the plane: it is Green's theorem. Take n = 3 with the divergence reading: it is the Gauss theorem; with the curl reading on a surface: it is classical Stokes. Crucially it is also the engine of de Rham cohomology: pairing closed forms with cycles is well-defined precisely because of Stokes (changing the form by an exact piece, or the cycle by a boundary, changes the integral by zero). A caution: the orientations must match the induced-boundary convention, or signs come out wrong; and compact support (or compactness of M) is needed so the integrals converge.
Green's theorem as Stokes: on a planar region M with boundary curve partial M, take omega = P dx + Q dy. Then d omega = (Q_x - P_y) dx^dy, and Stokes says the integral over M of (Q_x - P_y) dx dy equals the integral over partial M of P dx + Q dy. Setting P = 0, Q = x gives the area of M as the line integral of x dy around its boundary — a planimeter computes area by tracing only the boundary, which is Stokes in action.
One formula, integral of d omega over M equals integral of omega over partial M, contains the fundamental theorem, Green, Gauss, and classical Stokes.
Stokes' theorem is the precise duality between d (algebra) and partial (geometry): integral of d omega over M equals integral of omega over partial M is the adjointness 'd is dual to taking boundary'. Get the induced orientation on partial M right or every sign is wrong; the outward-normal-first convention is the standard one.