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Unformatted text preview: 4.3. STOKESâ€™S THEOREM 101 and d Î³ Î± = p X i = 1 âˆ‚Î³ Î±, i âˆ‚ x i dx 1 âˆ§ Â·Â·Â· âˆ§ dx p 14. Therefore, Z U Î± d Î³ Î± = p X i = 1 Z U Î± âˆ‚Î³ Î±, i âˆ‚ x i dx 1 âˆ§Â·Â·Â·âˆ§ dx p = p X i = 1 Z R p âˆ‚Î³ Î±, i âˆ‚ x i dx 1 âˆ§Â·Â·Â·âˆ§ dx p = . 15. Hence Z V Î± d ( Ï• Î± Î² ) = . 16. Also, since U Î± is disjoint from the boundary Z âˆ‚ V Ï• Î± Î² = . 17. Thus, for each chart disjoint from the boundary Z V Î± d ( Ï• Î± Î² ) = Z âˆ‚ V Ï• Î± Î² = . 18. Case II. Now, let us consider the halfopen charts V Î± at the boundary. 19. Let U Î± âŠ‚ R p be the halfopen sets in R p such that f Î± : U Î± â†’ V Î± be the local coordinate di ff eomorphisms. 20. Let W Î± = V Î± âˆ© âˆ‚ V and Y Î± = f 1 Î± ( W Î± ) . 21. Notice that for any point on Y Î± , x p = 0. 22. Then Z V Î± d ( Ï• Î± Î² ) = p X i = 1 Z R p âˆ‚Î³ Î±, i âˆ‚ x i dx 1 âˆ§ Â·Â·Â· âˆ§ dx p 23. We have Z âˆžâˆž âˆ‚Î³ Î±, i âˆ‚ x i dx i = for any i , p , and Z âˆž âˆ‚Î³ Î±, p âˆ‚ x p dx p = Î³ Î±, p x p = ....
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 Spring '10
 Wong
 Geometry, Manifold, dx, yÎ±, Ï•Î± Î²

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