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lecture37

# lecture37 - Lecture 37 6.10 Integration on Smooth Domains...

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Unformatted text preview: Lecture 37 6.10 Integration on Smooth Domains Let X be an oriented n-dimensional manifold, and let ω ∈ Ω c n ( X ). We defined the integral ω, (6.136) X but we can generalize the integral ω, (6.137) D for some subsets D ⊆ X . We generalize, but only to very simple subsets called smooth domains (essentially manifolds-with-b 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 φ : U → V of X at p such that φ ( U ∩ H n ) = V ∩ D . Definition 6.44. The map φ is a parameterization of D at p . Note that φ : 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 ψ = φ | U b . Then ψ : U b V b is a diffeomorphism. The set U b is an open set → in R n − 1 , and ψ 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 = φ (the empty set)....
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lecture37 - Lecture 37 6.10 Integration on Smooth Domains...

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