lecture38 - Lecture 38 We begin with a review from last...

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Lecture 38 We begin with a review from last time. Let X be an oriented manifold, and let D X be a smooth domain. Then Bd ( D ) = Y is an oriented ( n 1)-dimensional manifold. We defined integration over D as follows. For ω Ω n c ( X ) we want to make sense of the integral ω. (6.161) D We look at some special cases: Case 1: Let p Int D , and let φ : U V be an oriented parameterization of X at p , where V Int D . For ω Ω c n ( X ), we define ω = ω = φ ω = φ ω. (6.162) D V U R n This is just our old definition for ω. (6.163) V Case 2: Let p Bd ( D ), and let φ : U V be an oriented parameterization of D at p . That is, φ maps U H n onto V D . For ω Ω n c ( V ), we define ω = φ ω. (6.164) D H n We showed last time that this definition does not depend on the choice of parameter- ization. General case: For each p Int D , let φ : U p V p be an oriented parameterization of X at p with V p Int D . For each p Bd ( D ), let φ : U V p be and oriented p
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