This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 μ and v ν be positivelyoriented coordinates on U and V , and the di ﬀ 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 α ∈ Λ p be a pform on M α = 1 p ! α i 1 ... i p ( x ) dx i 1 ∧ ··· ∧ dx i p di ﬀ geom.tex; April 12, 2006; 17:59; p. 93...
View
Full Document
 Spring '10
 Wong
 Geometry, Integrals, Stokes' theorem, PARAMETRIZED SUBSET

Click to edit the document details