21-356 Lecture 38

E2 there exist a ball b x0 r and a function g c 1 b x0

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Unformatted text preview: y x0 ! E2 there exist a ball B (x0 , r) and a function g ! C 1 (B (x0 , r)) such that with ,g (x) &= 0 for all x ! B (x0 , r) + # U , such that B (x0 , r) + U = {x ! B (x0 , r) : g (x) < 0} , B (x0 , r) \ U = {x ! B (x0 , r) : g (x) > 0} , B (x0 , r) + # U = {x ! B (x0 , r) : g (x) = 0} . Note that the radius of the ball and the function g depend on x0 . Example 178 Let’s calculate the outward flux of the function f (x, y, z ) := (0, yz, x) across the boundary of the region $ % U := (x, y, z ) ! R3 : x2 + y 2 < z 2 , x2 + y 2 + z 2 < 2y, z > 0 . Note that U is not a regular open set (why?) but it satisfies the hypotheses...
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This document was uploaded on 03/31/2014.

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