Converting polar coordinate expression to cartesian coordinates

Question

An intermediate step in my analysis requires me to work in polar coordinates, but I would like to convert the results back into cartesian coordinates. The conversion is very simple but tedious for complex expressions. I expected to use substitutions, but it isn't panning out for me

polarf[{r_, phi_}] := r^6 Sin[phi]^2 Cos[phi]^2
cartesianf[{k1_, k2_}] := 
 polarf[{r, phi}] /. r^2 Sin[phi]^2 -> k2^2 /. 
   r^2 Cos[phi]^2 -> k1^2 /. r^2 -> k1^2 + k2^2
cartesianf[{k1, k2}]

gives me back my original expression:

r^6 Cos[phi]^2 Sin[phi]^2

I'm pretty sure there is a way to get Mathematica to spit out

(k1^2 + k2^2) k1^2 k2^2

or something similar. But how? My searches of Google, Mathematica documentation, and this site have not turned up an answer.

Answer 1

You obviously misunderstand replacement rules. Replacement rules work on the structure level, and don't care about the mathematical meaning (they do take into accounts attributes like Orderless, however). In your case, you are saying with your first replacement rule that whereever your result contains the structure (more exactly, the pattern) r^2 Sin[phi]^2 (that is, Times[Power[r, 2], Power[Sin[phi], 2]]), you want that replaced with the expression k2^2. However the result of your function does not contain this structure. It contains r^6 Sin[phi]^2 which doesn't match your pattern because 6 is not 2. For the same reason, your second replacement rule doesn't do anything, either.

What does work is

polarf[{r, phi}] /. { Cos[phi] -> k1/r, Sin[phi] -> k2/r } /.
                    { r -> Sqrt[k1^2 + k2^2] }

This works because the expression generated by polarf, r^6 Cos[phi]^2 Sin[phi]^2, does contain Cos[phi] and Sin[phi], which in the first step get replaced by k1/r and k2/r. Thus you get r^6 (k1/r)^2 (k2/r)^2 which gets further evaluated to r^2 k1^2 k2^2. Now the last replacement rule kicks in, replacing r by Sqrt[k1^2 + k2^2]. Thus you get Sqrt[k1^2 + k2^2]^2 k1^2 k2^2 which evaluates to (k1^2 + k2^2) k1^2 k2^2, which is the expression you want.