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21-356 Lecture 38 - = A"U.k f 1 dHN#1 where we have used...

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Remark 176 In physics B " U f ( x ) · $ ( x ) d H N # 1 ( x ) represents the outward flux of a vector field f across the boundary of a region U . If E ) R N and f : E * R N is di " erentiable, then f is called a divergence-free field or solenoidal field if div f = 0 . Thus for a smooth solenoidal field, the outward flux across the boundary of a regular set U is zero. 104
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Corollary 177 The theorem continues to hold if U ) R N is open, bounded, and its boundary consists of two sets E 1 and E 2 , where E 1 is a closed set contained in the finite union of compact surfaces of class C 1 and dimension less than N ' 1 , while for every x 0 ! E 2 there exist a ball B ( x 0 , r ) and a function g ! C 1 ( B ( x 0 , r )) such that with , g ( x ) & = 0 for all x ! B ( x 0 , r ) + # U , such that B ( x 0 , r ) + U = { x ! B ( x 0 , r ) : g ( x ) < 0 } , B ( x 0 , r ) \ U = { x ! B ( x 0 , r ) : g ( x ) > 0 } , B ( x 0 , r ) + # U = { x ! B ( x 0 , r ) : g ( x ) = 0 } . Note that the radius of the ball and the function g depend on x 0 . 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 ) ! R 3 : x 2 + y 2 < z 2 , x 2 + y 2 + z 2 < 2 y, z > 0 % .
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