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 Z G ( V ) α = Z V G * α = 1 p ! Z V α i 1 ... i p ( G ( v )) x i 1 v 1 ··· x i p v p dv 1 ··· dv p = 1 p ! Z 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 ! Z U α i 1 ... i p ( F ( u )) x i 1 u 1 ··· x i p u p du 1 ··· du p = Z 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 Z ψ ( U ) α = Z U ψ * α = Z U ( F * ϕ * ) α = Z U F * ( ϕ * α ) = Z F ( U ) ϕ * α Let S = F ( U ) be an oriented subset of

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## This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.

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

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