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differential geometry w notes from teacher_Part_45

differential geometry w notes from teacher_Part_45 -...

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Chapter 4 Integration of Di ff erential Forms 4.1 Integration over a Parametrized Subset 4.1.1 Integration of n -Forms in R n Let U R n be a closed ball in R n and u i , i = 1 , . . . , n be the Cartesian coordinates in R n . Let f : U R be a continuous real-valued function on U . Then the integral of f over U is the multiple integral U f = U f ( u ) du 1 · · · du n . Let o be an orientation of U so that o ( u ) = + 1 if the coordinate basis of cov- ectors ( du 1 , . . . , du n ) has the same orientation as o and o ( u ) = - 1 otherwise. Let α = f ( u ) du 1 ∧ · · · ∧ du n be an n -form. Then the integral of α over U is defined by U α = o ( u ) U f ( u ) du 1 . . . du n . The integral of the form α over U reverses sign if the orientation of U is reversed. 89
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90 CHAPTER 4. INTEGRATION OF DIFFERENTIAL FORMS 4.1.2 Integration over Parametrized Subsets Let M be an n -dimensional manifold with local coordinates x i , i = 1 , . . . , n . Let 0 p n and U be an oriented region in R p with orientation o and coordinates u μ , μ = 1 , . . . , p . Let F : U M be a smooth map given locally by x i = F i ( u ) . Then the image F ( U ) M of the set U is called a p -subset of M and the collection ( U , o , F ) is called an oriented parametrized p -subset of
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