quiz11' - div( F G ) = G curl F-F curl G . Letting F = f...

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Quiz 11 Solutions 1. Let f and g be scalar functions of three variables with continuous second-order partial derivatives. Compute div( f × ∇ g ). To get full credit on this problem, you must simplify completely. (This is exercise 28 in section 16.5. Exercises 23-29 in that section asked you to prove various identities involving divergence, curl, and the Laplacian. If you can remember what any of those exercies other than number 28 say, you may use them without proving them.) If you don’t remember any of those other exercises, you can compute the divergence as follows. The conditions of Clairaut’s Theorem are satisfied for f and g , and we use that theorem at the end of the computation to put all the subscripts in alphabetical order. div( f × ∇ g ) = div ± ± ± ± ± ± i j k f x f y f z g x g y g z ± ± ± ± ± ± = ∂x ( f y g z - f z g y ) + ∂y ( f z g x - f x g z ) + ∂z ( f x g y - f y g x ) = f y g zx + f yx g z - f z g yx - f zx g y + f z g xy + f zy g x - f x g zy - f xy g z + f x g yz + f xz g y - f y g xz - f yz g x = f y g xz + f xy g z - f z g xy - f xz g y + f z g xy + f yz g x - f x g yz - f xy g z + f x g yz + f xz g y - f y g xz - f yz g x = 0 . However, an easier way to do this is to remember exercise 27, which says
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Unformatted text preview: div( F G ) = G curl F-F curl G . Letting F = f and G = g , this gives us that div( f g ) = 0, since the curl of any gradient eld is . 2. Find the ux of the vector eld F ( x, y, z ) = x ( x 2 + y 2 + z 2 ) 3 / 2 i + y ( x 2 + y 2 + z 2 ) 3 / 2 j + z ( x 2 + y 2 + z 2 ) 3 / 2 k through the surface x 2 + y 2 + z 2 = 1, oriented outward. Note that F always points in the same direction as the normal to this surface. So, at any point on the surface, F ( x, y, z ) n = | F ( x, y, z ) | | n | = s x 2 ( x 2 + y 2 + z 2 ) 3 + y 2 ( x 2 + y 2 + z 2 ) 3 + z 2 ( x 2 + y 2 + z 2 ) 3 = s 1 ( x 2 + y 2 + z 2 ) 2 = 1 . Therefore, if we call the given surface D , the ux integral is given by Z D F d S = Z D F n dS = Z D dS, which is the surface area of D . This is 4 . 1...
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