How do I express an equation in Polar coordinates as a Cartesian one.

[JorgeM]

TL;DR Summary

I need to convert an equation that is on polar coordinates into a cartesian but as soon I start doing that I got confused and I'm not really shure about what to do.

I got a polar function.

$$ \psi = P(\theta )R(r) $$

When I calculate the Laplacian:
$$ \ \vec \nabla^2 \psi = P(\theta)R^{\prime\prime}(r) + \frac{P(\theta)R^{\prime}(r)}{r} + \frac{R(r)P^{\prime\prime}(\theta)}{r^{2}}
$$

Now I need to convert this one into cartesian coordinates and then it results very difficult to me because I know how to convert simple equations using the simple relations:$$ x = r Cos(\theta ) $$$$ y = r Sin(\theta ) $$

$$ r= \sqrt{x^{2}+y^{2}} $$

$$ \theta= arcTan( \frac{y}{x}) $$I can't figure out how to use this relations in order to replace them in my functions R and P since there are first and second derivatives of a functions dependent on x and y, so I can not just replace the relations (At least not directly).

What I need to do is to express the Laplacian of psi in cartesian coordinates.
Is there a way just to replace in them as :

$$ P(\theta) = F(x,y)$$
$$ P^{\prime}(\theta) = G(x,y)$$

$$ P^{\prime\prime}(\theta) = H(x,y)$$

Or how should I try?

Thanks A lot for your help

 

[mfb]

If you don't know P and R, the best you can do is setting up something like $\displaystyle \frac{\partial P(\theta(x,y))}{\partial \theta}$ and then applying the chain rule. It will still look quite awkward.