Stokes' theorem

Let $M$ be an oriented smooth $n$-manifold with (possibly empty) boundary $\partial M$, and let $\iota:\partial M \hookrightarrow M$ be the inclusion. If $\alpha \in \Omega^{n-1}(M)$ is smooth and either compactly supported in the interior of $M$ or $M$ is compact with $\alpha$ smooth up to the boundary, then Stokes’ theorem states

Orientation convention

The boundary $\partial M$ is oriented by the outward-normal-first rule: a basis $(v_1,\dots,v_{n-1})$ of $T_p(\partial M)$ is positively oriented if $(\nu, v_1,\dots,v_{n-1})$ is a positively oriented basis of $T_p M$, where $\nu$ points outward (or is a chosen outward-pointing normal vector field along $\partial M$).

Key special cases

Stokes’ theorem unifies several classical theorems:

• Fundamental theorem of calculus ($n=1$): for $\alpha=f$ (a $0$-form) on $[a,b]$, $\int_{[a,b]} df \;=\; \int_{\{b\}} f - \int_{\{a\}} f \;=\; f(b)-f(a).$
• Green’s theorem and the divergence theorem arise from choosing appropriate $(n-1)$-forms corresponding to vector fields.
• The classical Kelvin–Stokes theorem in $\mathbb{R}^3$ is the case $n=2$ applied to a surface with boundary.

Useful corollaries

• If $\partial M=\varnothing$, then for any $\alpha \in \Omega^{n-1}(M)$, $\int_M d\alpha = 0.$
• If $\beta$ is a closed $k$-form ($d\beta=0$), then $\int_{\partial C} \beta = 0$ for any $(k+1)$-chain $C$ for which the integrals make sense, since $\int_{\partial C}\beta=\int_C d\beta$.

For an important class of closed $2$-forms, see symplectic manifolds .