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Unformatted text preview: Lecture 37 6.10 Integration on Smooth Domains Let X be an oriented ndimensional manifold, and let c n ( X ). We defined the integral , (6.136) X but we can generalize the integral , (6.137) D for some subsets D X . We generalize, but only to very simple subsets called smooth domains (essentially manifoldswithb oundary). The prototypical smooth domain is the half plane: H n = { ( x 1 , . . . , x n ) R n : x 1 } . (6.138) Note that the boundary of the half plane is Bd ( H n ) = { ( x 1 , . . . , x n ) R n : x 1 = 0 } . (6.139) Definition 6.43. A closed subset D X is a smooth domain if for every point p Bd ( D ), there exists a parameterization : U V of X at p such that ( U H n ) = V D . Definition 6.44. The map is a parameterization of D at p . Note that : U H n V D is a homeomorphism, so it maps boundary points to boundary points. So, it maps U b = U Bd ( H n ) onto V b = V Bd ( D ). Let =  U b . Then : U b V b is a diffeomorphism. The set U b is an open set in R n 1 , and is a parameterization of the Bd ( D ) at p . We conclude that Bd ( D ) is an ( n 1)dimensional manifold. (6.140) Here are some examples of how smooth domains appear in nature: Let f : X R be a C map, and assume that f 1 (0) C f = (the empty set)....
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