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differential geometry w notes from teacher_Part_47

differential geometry w notes from teacher_Part_47 - 93 4.1...

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4.1. INTEGRATION OVER A PARAMETRIZED SUBSET 93 Then G ( V ) α = V G * α = 1 p ! V α i 1 ... i p ( G ( v )) x i 1 v 1 · · · x i p v p dv 1 · · · dv p = 1 p ! U α i 1 ... i p ( G ( H ( u ))) x i 1 v 1 · · · x i p v p ( v 1 , . . . , v p ) ( u 1 , . . . , u p ) du 1 · · · du p = 1 p ! U α i 1 ... i p ( F ( u )) x i 1 u 1 · · · x i p u p du 1 · · · du p = U α Thus, the integral is independent of the parametrization of a p -subset. 4.1.6 Integrals and Pullbacks Let M be an n -dimensional manifold and W be an r -dimensional manifold. Let ϕ : M W be a smooth map. Let U R p be an oriented region in R n and F : U M be an oriented parametrized p -subset of M . Then ψ = ϕ F : U W is an oriented parametrized p -subset of W . Let α Λ p W be a p -form on W . Then ψ ( U ) α = U ψ * α = U ( F * ϕ * ) α = U F * ( ϕ * α ) = F ( U ) ϕ * α Let S = F ( U ) be an oriented subset of
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