21-356 Lecture 38

A sin a z dxdydz u 0 a 0 0 a a 0 0 a 0 a i 2r

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Unformatted text preview: , r < sin !, r < z < 2r sin ! ' r2 . 105 Hence, by changing variables A A ! "A sin * "A z dxdydz = U 0 = A 0 ! 0 = A ! A 0 ! 0 = A 0 "A ! I 2r sin * #r 2 z dz r sin * 0 0 = "A & sin * I ' z2 2 Jz=&2r sin *#r2 z =r 2 r sin ! ' r 3 ( # r dr r dr # # # d! d! dr d! Jr=sin * J A !I r3 r4 1 1 sin ! ' d! = sin4 ! ' sin4 ! d! 3 4 r=0 3 4 0 1 1 sin4 ! d! = ". 12 32 Corollary 179 (Integration by Parts) Let U ) RN be an open, bounded, regular set and let f : U * R and g : U * R be such that f and g are bounded and continuous in U and there exist the partial derivat...
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