differential geometry w notes from teacher_Part_46

# differential geometry w notes from teacher_Part_46 - 1 dv p...

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4.1. INTEGRATION OVER A PARAMETRIZED SUBSET 91 4.1.3 Line Integrals Let U = [ a , b ] R be an interval. Then a map F : U M deﬁnes an oriented curve C = F ( U ) in M x i = F i ( t ) . Let α be a 1-form in M α = α i ( x ) dx i . Then the integral of α over C is called the line integral . In more detail Z C α = Z b a α " F * ± d dt !# = Z b a α i ( x ( t )) dx i dt dt . 4.1.4 Surface Integrals Let U R 2 be an oriented region in the plane, for example, U = [ a , b ] × [ c , d ]. Then a map F : U M deﬁnes an oriented parametrized surface S = F ( U ) in M x i = F i ( u 1 , u 2 ) . Let α be a 2-form in M α = 1 2 α i j ( x ) dx i dx j . Then the integral of α over C is called the surface integral . In more detail Z S α = Z U α " F * ± d du 1 ! , F * ± d du 2 !# du 1 du 2 = Z U 1 2 α i j ( x ( u )) dx i du 1 dx j du 2 du 1 du 2 . Example in R 3 . di geom.tex; April 12, 2006; 17:59; p. 92

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92 CHAPTER 4. INTEGRATION OF DIFFERENTIAL FORMS 4.1.5 Independence of Parametrization Let U , V R p be regions in R p with coordinates u μ , μ = 1 , . . . , p , and v ν , ν = 1 , . . . , p , respectively. Let H : U V be an di eomorphism so that V = H ( U ) deﬁned by v μ = H μ ( u ) . Then for any function f : V R there holds a formula for the change of variables in multiple integrals Z V f ( v ) dv
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Unformatted text preview: 1 . . . dv p = Z U f [ v ( u )] ± ± ± ± ± ± ∂ ( v 1 , . . . , v p ) ∂ ( u 1 , . . . , u p ) ± ± ± ± ± ± du 1 . . . du p • Let M be an n-dimensional manifold with n ≥ p . • Let U and V be oriented regions, u μ and v ν be positively-oriented coordi-nates on U and V , and the di ﬀ eomorphism H be orientation-preserving, that is, the Jacobian ∂ ( v 1 , . . . , v p ) ∂ ( u 1 , . . . , u p ) > is positive. • Let F : U → M and G : V → M be a smooth maps so that F = G ◦ H . • Then F ( U ) is an oriented parametrized p-subset of M and G is a reparametriza-tion of this subset x i = F i ( u ) = G i ( H ( u )) . • Let α ∈ Λ p be a p-form on M α = 1 p ! α i 1 ... i p ( x ) dx i 1 ∧ ··· ∧ dx i p di ﬀ geom.tex; April 12, 2006; 17:59; p. 93...
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differential geometry w notes from teacher_Part_46 - 1 dv p...

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