Stokes' Theorem on Manifolds

One theorem: integrate the derivative over the inside, integrate the form over the boundary.

The idea

Theorem (Generalized Stokes theorem).

Let $M$ be a compact oriented smooth $k$-manifold with boundary $\partial M$, the boundary carrying the induced orientation, and let $\omega$ be a smooth $(k-1)$-form on $M$. Then $\int_{M} d\omega = \int_{\partial M} \omega.$

The degrees match on both sides. The exterior derivative $d$ raises the degree of a form by one, so $d\omega$ is a $k$-form, integrable over the $k$-dimensional $M$; the boundary $\partial M$ has dimension $k-1$, so the $(k-1)$-form $\omega$ is integrable over it. Compactness keeps both integrals finite, and the boundary must carry the induced orientation, or the two sides differ by a sign.

The equation says that the integral of a derivative over a region is determined by the values on the region's boundary. For $k = 1$, $M = [a,b]$ and $\omega = f$ a $0$-form, the boundary is the two endpoints, which the induced orientation counts with opposite signs, and the theorem reads $\int_{a}^{b} f'\,dx = f(b) - f(a)$ — the Fundamental Theorem of Calculus. Green's theorem and the classical curl theorem are the case $k = 2$, for a plane region and for a surface in $\mathbb{R}^{3}$, and the divergence theorem is the case $k = 3$, so this one statement contains all four classical results.

Ways to work on it

• Walkthrough. The generalized Stokes theorem and how it specializes to the fundamental theorem of calculus and Green's theorem.
• Practice. Identify which classical integral theorem Stokes reduces to in a given setting.
• Hardest. Use Green's theorem to compute a boundary integral as an area.

Not sure where to start? Take the ten-question placement test.