- #1
Silviu
- 624
- 11
Hello! I am reading how to integrate on an orientable manifold. So we have ##f:M \to R## and an m-form (m is the dimension of M): ##\omega = h(p)dx^1 \wedge ... \wedge dx^m##, where ##h(p)## is another function on the manifold which is always positive as the manifold is orientable. The way integral is defined is like this: ##\int_{U_i} f\omega = \int_{\phi(U_i)}f(\phi^{-1}(x))h(\phi^{-1}(x))dx^1...dx^m##, where ##U_i## are the coordinate neighborhoods and ##\phi## is the mapping from M to ##R^m##. The definition makes sense, intuitively, by making an analogy with the surface or volume integrals. However I am not sure formally, how did the wedge product of ##\omega##, transformed into simple multiplication. Can someone explain this to me? Thank you!