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Unformatted text preview: 4.3. STOKESS THEOREM 101 and d = p X i = 1 , i x i dx 1 dx p 14. Therefore, Z U d = p X i = 1 Z U , i x i dx 1 dx p = p X i = 1 Z R p , i x i dx 1 dx p = . 15. Hence Z V d ( ) = . 16. Also, since U is disjoint from the boundary Z V = . 17. Thus, for each chart disjoint from the boundary Z V d ( ) = Z V = . 18. Case II. Now, let us consider the halfopen charts V at the boundary. 19. Let U R p be the halfopen sets in R p such that f : U V be the local coordinate di ff eomorphisms. 20. Let W = V V and Y = f 1 ( W ) . 21. Notice that for any point on Y , x p = 0. 22. Then Z V d ( ) = p X i = 1 Z R p , i x i dx 1 dx p 23. We have Z  , i x i dx i = for any i , p , and Z , p x p dx p = , p x p = ....
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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.
 Spring '10
 Wong
 Geometry

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