The equations F, G express the Cartesian coordinates x,y in terms of the cylindrical coordinates r, θ. But the question asks for how to do it the other way round: express the cylindrical coordinates in terms of the Cartesian coordinates.

The third coordinate z (cylindrical) is the same as z (Cartesian), so there is no problem finding z. There is also no problem finding r, because it is given by the formula . So the question is really about discussing whether θ can be reconstructed in a neighbourhood of the point (x,y,z).

Ah, I see, you're trying to use a more sophisticated approach, inverting a Jacobian matrix. (I should have realised that, from the title of the thread.) In that case, your third equation should be . The two z's are conceptually different, and the relation between them should result in a 1 somewhere in the Jacobian matrix.

In this problem, the sophisticated method can obscure the simple geometry of the transformation, and a naive approach may give more insight. Maybe that is why part b. of the question asks you to think about the geometric meaning of your answer.