Stokes' theorem is the global identity behind the fundamental theorem of calculus, Green's theorem, the divergence theorem, and the classical curl theorem. Its power is that the same exterior derivative governs all dimensions.
Theorem (Stokes). Let \(M\) be an oriented smooth \(n\)-manifold with boundary and let \(\omega\in\Omega_c^{n-1}(M)\). Then \[ \int_M\mathrm{d}\omega=\int_{\partial M}\omega, \] where \(\partial M\) has the outward-normal-first orientation.
Reduction to a half-space chart
Proof. Choose an oriented partition of unity subordinate to interior charts and boundary charts. By linearity it suffices to prove the result when \(\omega\) is supported in one chart. In an interior chart, write \[ \omega=\sum_{i=1}^n(-1)^{i-1}f_i\, \mathrm{d}x^1\wedge\cdots\widehat{\mathrm{d}x^i}\cdots \wedge\mathrm{d}x^n. \] Then \(\mathrm{d}\omega=(\sum_i\partial_if_i)\,\mathrm{d}x^1\wedge \cdots\wedge\mathrm{d}x^n\), whose integral is zero by compact support. In a boundary chart modeled on \(\mathbb H^n=\{x^n\ge0\}\), all \(i<n\) terms again integrate to zero. The remaining term gives, by the one-dimensional fundamental theorem, \[ \int_{\mathbb H^n}\partial_nf_n\,\mathrm{d}x =-\int_{\mathbb R^{n-1}}f_n(x',0)\,\mathrm{d}x'. \] The induced boundary orientation contributes exactly this sign. Summing the partition pieces proves the theorem.
Classical integral theorems
Example (fundamental theorem). For \(M=[a,b]\) and a function \(f\), \[ \int_a^b\mathrm{d}f=f(b)-f(a). \]
Example (divergence theorem). On an oriented Riemannian manifold, let \(X\) be a vector field and set \(\omega=\iota_X\operatorname{vol}_g\). Since \[ \mathrm{d}\omega=(\operatorname{div}X)\operatorname{vol}_g, \] Stokes gives \[ \int_M\operatorname{div}X\,\mathrm{d}V_g =\int_{\partial M}\langle X,\nu\rangle\,\mathrm{d}A_g. \]
Consequences
Corollary. If \(M\) is compact and has no boundary, then \(\int_M\mathrm{d}\omega=0\). In particular, an exact top form cannot have nonzero integral.
If a closed curve \(\gamma\) bounds an oriented surface \(S\), then \[ \int_\gamma\alpha=\int_S\mathrm{d}\alpha. \] When no such surface exists, the obstruction is topological rather than a failure of calculus. This observation leads directly to homology and de Rham cohomology.
Manifolds with corners
For piecewise smooth domains, codimension-two faces occur twice with opposite induced orientations, so they cancel. The slogan \(\partial^2=0\) is the geometric counterpart of \(\mathrm{d}^2=0\).