- #1
JG89
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First let me write out the definition of a manifold given in my book:
Let k > 0. A k-manifold in \mathbb{R}^n of class C^r is a subspace M of \mathbb{R}^n having the following property: For each p \in M, there is an open set V \subset M containing p, a set U that is open in either \mathbb{R}^k or \mathbb{H}^k (upper half space), and a continuous bijection \alpha : U \rightarrow V such that 1) \alpha is of class C^r, 2) \alpha^{-1} : V \rightarrow U is continuous, 3) D\alpha(x) has rank k for each x \in U. The map \alpha is called a coordinate patch on M about p.
In my text I am reading the chapter on integrating a scalar map over a compact manifold. My question is this: Suppose M is a compact-manifold. As a subset of \mathbb{R}^n it is bounded. So instead of going through all the mess of defining a manifold and defining the integral of a continuous function f over a manifold, why not just integrate f over M as one usually would? Using Riemann sums in \mathbb{R}^n?
Surely this would give the same result as integrating f over M using the definition of integral over a manifold. So what's so special about using a manifold M for integration when we could just consider M as a regular bounded subset of Euclidean space and integrate it how we usually would?
In case you're wondering, here is the definition of the integral of a scalar map over a compact manifold M:
Let M be a compact k-manifold in \mathbb{R}^n, of class C^r. Let f: M \rightarrow \mathbb{R} be a continuous function. Suppose that \alpha_i: A_i \rightarrow M_i, for i = 1, ..., N, is a coordinate patch on M, such that A_i is open in \mathbb{R}^k and M is the disjoint union of the open sets M_1, M_2, ..., M_N of M and a set K of measure zero in M. Then \int_M f dV = \sum_{i = 1}^N \int_{A_i} f(\alpha_i) V(D \alpha_i).
Note that dV represents the integral with respect to volume and V(D \alpha_i) = \sqrt{det[(D\alpha_i)^{tr} D\alpha_i]}
Let k > 0. A k-manifold in \mathbb{R}^n of class C^r is a subspace M of \mathbb{R}^n having the following property: For each p \in M, there is an open set V \subset M containing p, a set U that is open in either \mathbb{R}^k or \mathbb{H}^k (upper half space), and a continuous bijection \alpha : U \rightarrow V such that 1) \alpha is of class C^r, 2) \alpha^{-1} : V \rightarrow U is continuous, 3) D\alpha(x) has rank k for each x \in U. The map \alpha is called a coordinate patch on M about p.
In my text I am reading the chapter on integrating a scalar map over a compact manifold. My question is this: Suppose M is a compact-manifold. As a subset of \mathbb{R}^n it is bounded. So instead of going through all the mess of defining a manifold and defining the integral of a continuous function f over a manifold, why not just integrate f over M as one usually would? Using Riemann sums in \mathbb{R}^n?
Surely this would give the same result as integrating f over M using the definition of integral over a manifold. So what's so special about using a manifold M for integration when we could just consider M as a regular bounded subset of Euclidean space and integrate it how we usually would?
In case you're wondering, here is the definition of the integral of a scalar map over a compact manifold M:
Let M be a compact k-manifold in \mathbb{R}^n, of class C^r. Let f: M \rightarrow \mathbb{R} be a continuous function. Suppose that \alpha_i: A_i \rightarrow M_i, for i = 1, ..., N, is a coordinate patch on M, such that A_i is open in \mathbb{R}^k and M is the disjoint union of the open sets M_1, M_2, ..., M_N of M and a set K of measure zero in M. Then \int_M f dV = \sum_{i = 1}^N \int_{A_i} f(\alpha_i) V(D \alpha_i).
Note that dV represents the integral with respect to volume and V(D \alpha_i) = \sqrt{det[(D\alpha_i)^{tr} D\alpha_i]}