MATH 599 Midterm: MATH 5B UCSB 5B Fall 06 5BpMidterm 2s
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1. (a) convert the point (1, 1, 1) from rectangular to cylindrical coordinates. To convert to cylindrical coordinates, all we need to do is change the x, y-coordinates to polar coordinates: r =px2 + y2 = 2, and = tan 1(y/x) = tan 1( 1) = /4, which is in the same quadrant as (1, 1). So (1, 1, 1) = ( 2, /4, 1) in cylindrical coordinates. (b) convert (2, /2, 2 /3) from spherical to rectangular coordinates. Solution. x = sin cos = 2 sin(2 /3) cos( /2) = 0, y = sin sin = 2 sin(2 /3) sin( /2) = 3, and z = cos = 2 cos(2 /3) = 1. So (2, /2, 2 /3) = (0, 3, 1) in rectangular coordinates: suppose z and w are functions of x and y given by the equations z = Find the jacobian matrix of the inverse mapping when (z, w) = (2, 0), and simplify your answer.
