Partition of Unity on Different Manifolds (Part 1. Introduction)

An application of partition of unityPartition of unity builds a bridge between local properties and global properties. A nice example is the Stokes’ theorem on manifolds. Suppose $\omega$ is a $(n-1)$-form with compact support on a oriented manifold $M$ of dimension $n$ and if $\partial{M}$ is given the induced orientation, then\int_M d\omega=\int_{\partial{M}}\omegaThis theorem can be proved in two steps. First, by Fubini’s theorem, one proves the identity on $\mathbb{R}^n$ and $\mathbb{H}^n$. ...

Submit