TheoremProved

Stokes' theorem

Statement

In classical vector calculus, let $S \subset \mathbb{R}^3$ be an oriented piecewise smooth surface with positively oriented boundary curve $\partial S$ and unit normal $\mathbf{n}$ , and let $\mathbf{F}$ be a continuously differentiable vector field on a neighborhood of $S$ . Then $\iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n}\,dS = \oint_{\partial S} \mathbf{F} \cdot d\mathbf{r}$ . In its general differential-form formulation, for any smooth oriented $n$ -dimensional manifold-with-boundary $M$ and any compactly supported smooth $(n-1)$ -form $\omega$ on $M$ , $\int_M d\omega = \int_{\partial M} \omega$ .

Why is it true?

Imagine tiling a surface $S$ with a fine mesh of tiny oriented loops, all circulating in the same counterclockwise sense. The curl $(\nabla \times \mathbf{F}) \cdot \mathbf{n}$ at each point measures the microscopic circulation of $\mathbf{F}$ around one tiny tile. When you add the circulations of all the tiles together, every interior edge is shared by two adjacent tiles that traverse it in opposite directions, so all interior contributions cancel out in pairs. Only the unshared outer edges along the boundary $\partial S$ survive, leaving the macro-circulation $\oint_{\partial S} \mathbf{F} \cdot d\mathbf{r}$ . Seen as $\int_M d\omega = \int_{\partial M} \omega$ , it says that integrating the exterior derivative of a form is dual to taking the boundary of the domain.

Proof sketch

Using a smooth partition of unity subordinate to an oriented atlas of $M$ , decompose $\omega = \sum_k \rho_k \omega$ so that it suffices to prove $\int_M d\omega = \int_{\partial M} \omega$ when $\omega$ is compactly supported inside a single coordinate chart modeled on $\mathbb{R}^n$ or the upper half-space $\mathbb{H}^n = \{x \in \mathbb{R}^n : x_n \ge 0\}$ . Writing $\omega = \sum_{i=1}^{n} f_i(x)\,dx_1 \wedge \dots \wedge \widehat{dx_i} \wedge \dots \wedge dx_n$ , its exterior derivative is $d\omega = \left(\sum_{i=1}^{n} (-1)^{i-1} \frac{\partial f_i}{\partial x_i}\right) dx_1 \wedge \dots \wedge dx_n$ . Applying Fubini's theorem and the one-variable fundamental theorem of calculus along each coordinate $x_i$ , the terms $i < n$ vanish because $f_i$ has compact support, while the $i = n$ term on $\mathbb{H}^n$ evaluates at the boundary $x_n = 0$ to match $\int_{\partial \mathbb{H}^n} \omega$ with its induced orientation.

Topics that use this theorem

Vector calculus Smooth manifolds Differential geometry

Related theorems

Divergence theorem (Gauss–Ostrogradsky theorem)Fundamental theorem of calculus Cauchy's integral theorem Gauss–Bonnet theorem

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Michael Spivak (1965). Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus
Victor J. Katz (1979). The History of Stokes' Theorem · DOI:10.1080/0025570X.1979.11976770