differential geometry w notes from teacher_Part_51

differential geometry w notes from teacher_Part_51 - 4.3...

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Unformatted text preview: 4.3. STOKES’S THEOREM 101 and d γ α = p X i = 1 ∂γ α, i ∂ x i dx 1 ∧ ··· ∧ dx p 14. Therefore, Z U α d γ α = p X i = 1 Z U α ∂γ α, i ∂ x i dx 1 ∧···∧ dx p = p X i = 1 Z R p ∂γ α, i ∂ x i dx 1 ∧···∧ dx p = . 15. Hence Z V α d ( ϕ α β ) = . 16. Also, since U α is disjoint from the boundary Z ∂ V ϕ α β = . 17. Thus, for each chart disjoint from the boundary Z V α d ( ϕ α β ) = Z ∂ V ϕ α β = . 18. Case II. Now, let us consider the half-open charts V α at the boundary. 19. Let U α ⊂ R p be the half-open sets in R p such that f α : U α → V α be the local coordinate di ff eomorphisms. 20. Let W α = V α ∩ ∂ V and Y α = f- 1 α ( W α ) . 21. Notice that for any point on Y α , x p = 0. 22. Then Z V α d ( ϕ α β ) = p X i = 1 Z R p ∂γ α, i ∂ x i dx 1 ∧ ··· ∧ dx p 23. We have Z ∞-∞ ∂γ α, i ∂ x i dx i = for any i , p , and Z ∞ ∂γ α, p ∂ x p dx p =- γ α, p x p = ....
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differential geometry w notes from teacher_Part_51 - 4.3...

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