Question: Suppose (u, v)=(u(x, y), v(x, y)) is a change of variables from the Cartesian (x, y) coordinate system to some other (u, v) coordinate system. Let x = (x, y)T and z = (u(x), v(x))T.
(a) Calculate the Jacobian for the rectangular coordinate system obtained by rotating the (x, y) system through an angle of θ.
(b) Calculate the Jacobian for the transformation from the (x, y) system to polar coordinates.