21-356 Lecture 22

T h t dt a b a a d c f h t

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Unformatted text preview: eal Analysis notes, A a b ; ; f (" (h (t))) |h! (t)| ;" ! (h (t)); dt = = A b a A d c ; ; f (" (h (t))) ;" ! (h (t)); h! (t) dt ; ; f (" (, )) ;" ! (, ); d, . On the other hand, if h is strictly decreasing, then h! % 0, h (a) = d, and h (b) = a. In this case by the change of variables in the Real Analysis notes, A a b ; ; f (" (h (t))) |h (t)| ;" ! (h (t)); dt = ' ! =' = A c 57 A b Aac d d ; ; f (" (h (t))) ;" ! (h (t)); h! (t) dt ; ; f (" (, )) ;" ! (, ); d, ; ; f (" (, )) ;" ! (, ); d, . Remark 93 In particular, if # is a regular curve and " : [0;L (# )] ; RN is , * the parametric representation obtained using arclength, then ;" ! (, ); = 1 for all but finitely many , . Hence, A f ds = ! A L(! ) f (" (, )) d, . 0 The following properties are left as an exercise. Proposition 94 Let # be a piecewise C 1 curve and let f, g : E * R be two continuous functions, where E contains the range of # . Then (i) for...
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