15_2_9B - z-axis intersect the ellipsoids x 2 + y 2 + 2 z 2...

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The field lines of F satisfy dx 2 x z = dy 2 y z = dz - x 2 + y 2 z 2 . From the first equation, dx x = dy y , so y = Ax for an arbitrary constant A . Therefore dx 2 x = z dz - ( x 2 + y 2 ) = z dz - x 2 ( 1 + A 2 ) , so - ( 1 + A 2 ) x dx = 2 z dz . Hence 1 + A 2 2 x 2 + z 2 = B 2 , or x 2 + y 2 + 2 z 2 = B , where B is a second arbitrary constant. The field lines of F are the ellipses in which the vertical planes containing the
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Unformatted text preview: z-axis intersect the ellipsoids x 2 + y 2 + 2 z 2 = B . These ellipses are orthogonal to all the equipotential surfaces of F ....
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