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Unformatted text preview: 4.2. INTEGRATION OVER MANIFOLDS 97 4.2.2 Integration over Submanifolds A manifold M is compact if every open cover of M has a finite subcover. Thus, very compact manifold has a finite atlas. A subset of R n is compact if it is closed and bounded. Let V be a pdimensional compact oriented manifold. Let { U } N = 1 be a finite atlas of V . Let each U be positively oriented. Let { } be a partition of unity on V . Let be a pform over V . The integral of over V is defined by Z V = N X = 1 Z U This integral does not depend on the atlas and the partition of unity. Now, let V be a pdimensional compact oriented submanifold of an n dimensional manifold M described by the inclusion map i : V M . Let p M be a pform on M . The integral of a pform on M over V M is defined by Z V = Z V i * ....
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 Spring '10
 Wong
 Geometry

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