Exterior derivatives and Stokes' theorem

For a smooth function f, the exterior derivative is its differential df. Extend the operation to forms using linearity, a graded product rule, and the fact that differentiating coordinate differentials gives zero. If a one-form on the plane is Pdx+Qdy, its derivative is $(\partial_x Q-\partial_y P)dx\wedge dy$. The minus sign comes from moving dy past dx, not from a separate convention invented for planar curl. Exterior differentiation is intrinsic and commutes with smooth pullback.

Applying d twice gives zero. In coordinates the mixed second derivatives cancel by antisymmetry for smooth coefficients. A form ω with dω=0 is called closed; a form equal to dη is called exact. Every exact form is closed, but the reverse implication is global and can fail. The Poincare lemma supplies a local converse on sufficiently small coordinate neighborhoods, or a global converse for positive-degree closed forms on a star-shaped open subset of Euclidean space.

For a compact oriented smooth n-manifold M with boundary and a smooth (n−1)-form ω defined up to the boundary, Stokes' theorem identifies the integral of dω over M with the integral of ω over its boundary. A compact-support version handles appropriate noncompact settings. The boundary orientation is induced by the outward-normal-first convention: an outward transverse vector followed by a positively oriented boundary basis must give the orientation of M. No Riemannian unit normal is required to define this convention.

For the standard dx∧dy orientation of a planar region, this convention makes the outer boundary counterclockwise and inner-hole boundaries clockwise. Reversing the orientation of the region reverses both sides of Stokes' theorem. If the region is closed and has no boundary, the integral of every exact top-degree form is zero. Green's theorem, the classical curl theorem, and the divergence theorem emerge after translating their vector fields into forms, sometimes using a metric to make that translation.

On the punctured plane, the one-form $\alpha=(-y\,dx+x\,dy)/(x^2+y^2)$ is smooth and closed. Restricted to an angular coordinate patch it is dθ. Around the positively oriented unit circle, however, its integral is 2π. If α were the differential of a globally defined real-valued smooth function, its integral around every closed curve would be zero. The nonzero circulation therefore proves that local angle functions cannot be assembled into one global real angle.

There is no contradiction with Stokes' theorem. The filled disk is not contained in the punctured domain where α is smooth. Applying the theorem to an annulus instead includes an inner boundary whose opposite orientation cancels the outer circulation. This is a reliable way to debug apparent paradoxes: check smoothness on the entire integration region, include every boundary component, and state orientations before simplifying integrals. A suggestive arrow plot cannot replace these domain checks.