differential geometry w notes from teacher_Part_49

# differential geometry w notes from teacher_Part_49 - 4.2...

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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 p-dimensional compact oriented manifold. • Let { U α } N α = 1 be a finite atlas of V . • Let each U α be positively oriented. • Let { ϕ α } be a partition of unity on V . • Let β be a p-form over V . • The integral of β over V is defined by Z V β = N X α = 1 Z U α ϕ α β • This integral does not depend on the atlas and the partition of unity. • Now, let V be a p-dimensional compact oriented submanifold of an n- dimensional manifold M described by the inclusion map i : V → M . • Let β ∈ Λ p M be a p-form on M . • The integral of a p-form β on M over V ⊂ M is defined by Z V β = Z V i * β ....
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differential geometry w notes from teacher_Part_49 - 4.2...

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