Can anyone help with cylindrical polar coords?

[kala]

Homework Statement

This is a question in my classical mechanics book, and i am not very good with polar coordinates. I am suppose to fine r, phi, z in terms of x,y,z.
Basically I need to derive the cylindrical polar coordinates from the Cartesian coordinates.
The question specifically asks Find expressions for the unit vectors r,phi,z of cylindrical polar coordinates in terms of the Cartesian coordinates.

The Attempt at a Solution

So far I have drawn a picture of cylinder and labeled everything i could. Now I know z=z, that is no problem. I know that x=r*cos[theta] and y=r*sin[theta], I also know that r=sqrt[x^2+y^2] and i know that phi=arctan[y/x]. The only way that i know how to derive these is drawing a triangle and showing it, is there any other way, like actually deriving them?

 

[Irid]

I don't think there is any other way. The polar coordinates, in the first place, are defined by the drawing you have made, so in my opinion, there is no more "fundamental" way to arrive at the relations.

 

[turin]

You can construct a matrix of partial derivatives that convert from polar to Cartesian (the way you know how to do, obviously, from your post). Then, invert the matrix.