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Unformatted text preview: Chapter 4 Integration of Di ff erential Forms 4.1 Integration over a Parametrized Subset 4.1.1 Integration of n-Forms in R n Let U R n be a closed ball in R n and u i , i = 1 , . . . , n be the Cartesian coordinates in R n . Let f : U R be a continuous real-valued function on U . Then the integral of f over U is the multiple integral Z U f = Z U f ( u ) du 1 du n . Let o be an orientation of U so that o ( u ) = + 1 if the coordinate basis of cov- ectors ( du 1 , . . . , du n ) has the same orientation as o and o ( u ) =- 1 otherwise. Let = f ( u ) du 1 du n be an n-form. Then the integral of over U is defined by Z U = o ( u ) Z U f ( u ) du 1 . . . du n . The integral of the form over U reverses sign if the orientation of U is reversed. 89 90 CHAPTER 4. INTEGRATION OF DIFFERENTIAL FORMS 4.1.2 Integration over Parametrized Subsets Let M be an n-dimensional manifold with local coordinates x i , i = 1 , . . . , n ....
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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