Differential Geometry (9): Integration on Manifolds and Stokes' Theorem

Wed, 17 Nov 2021 09:00:00 +0000

In single-variable calculus, the fundamental theorem says that integrating a derivative over an interval equals the boundary difference: $\int_a^b f'(x)\,dx = f(b) - f(a)$ . The “boundary” of $$[a, b]$$ is the two-point set $\{a, b\}$ , with $$b$$ counted positively and $$a$$ negatively. The right-hand side is the integral of $$f$$ over this signed boundary. The left-hand side is the integral of the derivative over the interval. This is, in essence, every “fundamental theorem” you have ever met — Green’s theorem in the plane, the divergence theorem in three dimensions, the classical Stokes’ theorem on surfaces. They are all instances of one statement on manifolds: the integral of $d\omega$ over $$M$$ equals the integral of $\omega$ over $\partial M$ .

Stokes-Theorem on Chen Kai Blog

Differential Geometry (9): Integration on Manifolds and Stokes' Theorem