Generalised Stokes theorem

Statement

Let $\Omega$ be an oriented smooth manifold of dimension $n$ with boundary $\partial \Omega$, and give $\partial \Omega$ the induced orientation. Let $\omega$ be a smooth differential form of degree $n - 1$ on $\Omega$ with compact support, and let $d\omega$ be its exterior derivative. Then $$\int_\Omega d\omega = \int_{\partial \Omega} \omega.$$

Remarks

For $\Omega = [a, b]$ and a function $f$, the theorem is the fundamental theorem of calculus, $\int_a^b f'(x) \, dx = f(b) - f(a)$. The boundary point $a$ gets the negative orientation. For a curve and a 0-form, it is the gradient theorem. For a plane region and a 1-form, it is Green's theorem. For a surface in $\mathbb{R}^3$ and a 1-form, it is the classical Stokes' theorem. For a solid in $\mathbb{R}^3$ and a 2-form, it is the divergence theorem. The orientation and the compact support are needed so that both integrals are defined.

Explanations

L1 human

The integral of a derivative over a region equals the integral of the original form over the boundary of the region. One formula covers the fundamental theorem of calculus and the three integral theorems of vector calculus.