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Extra - i 1 If true justify and if false give a...

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i 1. If true, justify and if false, give a counterexample, or explain why. (a) Let f ( x, y, z ) = y - x . Then the line integral of f around the unit circle x 2 + y 2 = 1 in the xy plane is π , the area of the circle. Solution. This is false. The line integral of any gradient around a closed curve is zero. (b) Let F be a smooth vector field in space and suppose that the circulation of F around the circle of radius 1 centered at (0 , 0 , 0) and lying in the xy -plane, is zero. Then ( ∇ × F )(0 , 0 , 0) = 0. Solution. This is false. There are two reasons this is wrong. First, one has to have zero circulation for circles of arbitrarily small radius. Second, one has to have the circles in planes with arbitrary normal vectors. An explicit counterexample is F = y k , which has ∇ × F = i . The circulation around any circle in the xy -plane is zero, but the curl at (0 , 0 , 0) is not zero. (c) The center of mass of the region between the spheres x 2 + y 2 + z 2 = 4 and x 2 + y 2 + z 2 = 9 having mass density δ ( x, y, z ) = sin[ π (7 - x 2 - y 2 + 5 z )] lies somewhere on the z -axis between z = - 3 and z = 3. Solution. This is true. The mass density is symmetric about the z -axis (this is because the function δ depends on x and y only in the combination r 2 = x 2 + y 2 ) and so the center of mass lies on this axis of symmetry. On the other hand, the center
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