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Unformatted text preview: Lecture 26 We continue our study of forms with compact support. Let us begin with a review. Let U R n be open, and let = f I ( x 1 , . . . , x n ) dx I , (5.48) I where I = ( i 1 , . . . , i k ) is strictly increasing and dx I = dx i 1 dx i k . Then is compactly supported every f I is compactly supported . (5.49) By definition, supp f I = { x U : f I ( x ) = 0 } . (5.50) We assume that the f I s are C 2 maps. Notation. c k ( U ) = space of compactly supported differentiable kforms on U . (5.51) Now, let n c ( U ) defined by = f ( x 1 , . . . , x n ) dx 1 dx n , (5.52) where f 0 c ( U ). Then = f ( x 1 , . . . , x n ) dx 1 dx n . (5.53) R n R n Last time we proved the Poincare Lemma for open rectangles R in R n . We assumed that c n (Int R ). That is, we assumed that c n ( R n ) such that supp Int R . We showed that for such the following two conditions are equivalent: 1. R n = 0, 2. There exists a n 1 (Int R ) such that d = 0. c Definition 5.9. Whenever k ( U ) and = d for some k 1 ( U ), we say that is exact . Definition 5.10. Whenever k ( U ) such that d = 0, we say that is closed . Observe that n c ( U ) = d = 0 . (5.54) Now we prove the Poincare Lemma for open connected subsets of R n ....
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 Fall '04
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