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Unformatted text preview: Lecture 28 Let U, V be connected open sets of R n , and let f : U V be a diffeomorphism. Then +1 if f is orient. preserving, deg( f ) = (5.124) if f is orient. reversing. 1 We showed that given any n c ( V ), f = . (5.125) V U Let = ( x ) dx 1 dx n , where C ( V ). Then f i f = ( f ( x )) det x j ( x ) dx 1 dx n , (5.126) so, f i ( f ( x )) det dx = ( x ) dx. (5.127) x j V U Notice that f i f is orientation preserving det x j ( x ) > , (5.128) f i f is orientation reversing det x j ( x ) < . (5.129) So, in general, f i dx. (5.130) ( f ( x )) det ( x ) x j U As usual, we assumed that f C . Remark. The above is true for C 1 , a compactly supported continuous function. The proof of this is in section 5 of the Supplementary Notes. The theorem is true even if only f C 1 (the notes prove it for f C 2 ). Today we show how to compute the degree in general. Let U, V be connected open sets in R n , and let f : U V be a proper C map....
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This note was uploaded on 10/02/2010 for the course MAT unknown taught by Professor Unknown during the Fall '04 term at MIT.
- Fall '04