9 Integration

Definition 9.1 (Integration on Euclidean spaces) If \(A \subset \mathbb R^n\), and a smooth \(n\)-form \(\omega\) is defined on a neighborhood of \(A\), then with stardard coordinates, \(\omega = f dx^1 \wedge \dots \wedge dx^n\), and its integration on \(A\) is given by \[ \int_A \omega = \int_A f(x) dx^1 \cdots dx^n \]

Definition 9.2 (Integration on Manifolds) If \((U, \phi)\) is a chart of a manifold \(M\), and \(\omega\) is an \(n\)-form with compact support in \(U\), then the integral of \(\omega\) on \(U\) is defined as \[ \int_U \omega = \int_{\phi(U)} \left ((\phi)^{-1} \right)^* \omega \]

If \(\{(U, \phi)\}\) is an oriented atlas of \(M\), choose a partition of unity \(\{ \rho \}\) subordinate to the atlas. If \(\omega\) is an \(n\)-form defined on \(M\) with compact support, then \[ \omega = \sum \rho \omega \] is a finite sum, and the integral of \(\omega\) over \(M\) with respect to the current orientation is defined as \[ \int_M \omega = \sum \int_U \rho\omega \]

Theorem 9.1 (Stokes) If \(\omega\) is an \((n-1)\)-form on a manifold \(M\), then \[ \int_M d\omega = \int_{\partial M} i^*\omega \] where \(i: \partial M \to M\) is the natural inclusion.