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Math Help - Integrating over a Manifold

  1. #1
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    Integrating over a Manifold

    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 textbook 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]}
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  2. #2
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    Manifolds can be defined abstractly without being embedded into an ambient space. Also integral on a manifold can be defined without a metric, while a differential structure is enough. Read more on this topic you'll understand.
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  3. #3
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    I forgot to mention that I am only talking about manifolds that are subspaces of Euclidean space. I haven't learned about abstract manifolds yet.

    Does the answer that I want require you to speak about abstract manifolds? Or can you properly answer my question while only referring to Euclidean space?
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  4. #4
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    Also, what are useful (in physics, computer science, or even other parts of math) applications of abstract manifolds? I can see how sub-manifolds of Euclidean space would be useful, though at the moment they seem "redundant" to me.
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    Bump...
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