15_6_18 - F out of the box is 0. b) If is a sphere of...

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18. Let F = F 1 i + F 2 j + F 3 k be a constant vector field. a) If R is a rectangular box, we can choose the origin and coordinate axes in such a way that the box is 0 x a , 0 y b , 0 z c . On the faces x = 0 and x = a we have ˆ N = - i and ˆ N = i respectively. Since F 1 is constant, the total flux out of the box through these two faces is ZZ 0 y b 0 z c ( F 1 - F 1 ) dy dz = 0 . The flux out of the other two pairs of opposite faces is also 0. Thus the total flux of
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Unformatted text preview: F out of the box is 0. b) If is a sphere of radius a we can choose the origin so that has equation x 2 + y 2 + z 2 = a 2 , and so its outward normal is ˆ N = x i + y j + z k a . Thus the flux out of is 1 a ZZ ( F 1 x + F 2 y + F 3 z ) ds = , since the sphere is symmetric about the origin....
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