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Unformatted text preview: 1 . . . dv p = Z U f [ v ( u )] ( v 1 , . . . , v p ) ( u 1 , . . . , u p ) du 1 . . . du p Let M be an ndimensional manifold with n p . Let U and V be oriented regions, u and v be positivelyoriented coordinates on U and V , and the di eomorphism H be orientationpreserving, that is, the Jacobian ( v 1 , . . . , v p ) ( u 1 , . . . , u p ) > is positive. Let F : U M and G : V M be a smooth maps so that F = G H . Then F ( U ) is an oriented parametrized psubset of M and G is a reparametrization of this subset x i = F i ( u ) = G i ( H ( u )) . Let p be a pform on M = 1 p ! i 1 ... i p ( x ) dx i 1 dx i p di geom.tex; April 12, 2006; 17:59; p. 93...
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 Spring '10
 Wong
 Geometry, Integrals

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