differential geometry w notes from teacher_Part_48

differential geometry w notes from teacher_Part_48 - 95 4.2...

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4.2. INTEGRATION OVER MANIFOLDS 95 4.2 Integration over Manifolds 4.2.1 Partition of Unity Let M be a manifold, p 0 M be a point in M and ( U , x ) be a local coordinate chart about p 0 . Let x i = x i ( p ) be the local coordinates of the point p and x i 0 = x i ( p 0 ) be the local coordinates of the point p 0 . Let || x - x 0 || = v t n X i = 1 ( x i - x i 0 ) 2 A neighborhood of p 0 is a subset of M deﬁned by B ε ( p 0 ) = { p M | || x - x 0 || < ε } . Every neighborhood of a point in M is an open set in M . Let A M be a subset of M . A point p M is called an accumulation point (or a limit point ) of A if every neighborhood of p contains at least one point in A other than p . A subset of M is closed if and only if it contains all of its limit points. A closure of A , denoted by ¯ A , is a set obtained by adding to A all its accumulation points. A closure of any set is a closed set. A function

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differential geometry w notes from teacher_Part_48 - 95 4.2...

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