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lecture28

# lecture28 - Lecture 28 Let U V be connected open sets of Rn...

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�� 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 ω Ω n c ( V ), f ω = ω. (5.125) ± V U Let ω = φ ( x ) dx 1 ∧ · · · ∧ dx n , where φ ∈ C 0 ( V ). Then ∂f i f ω = φ ( f ( x )) det ∂x j ( x ) dx 1 ∧ · · · ∧ dx n , (5.126) so, ∂f i φ ( f ( x )) det dx = φ ( x ) dx. (5.127) ∂x j ± V U Notice that ∂f i f is orientation preserving det ∂x j ( x ) > 0 , (5.128) ∂f i f is orientation reversing det ∂x j ( x ) < 0 . (5.129) So, in general, ∂f i dx. (5.130) φ ( f ( x )) det ( x ) ∂x j U As usual, we assumed that f ∈ C . Remark. The above is true for φ ∈ C 0 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.

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lecture28 - Lecture 28 Let U V be connected open sets of Rn...

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