Transform the equation

$\displaystyle u_t=k(u_{xx}+u_{yy})$

to polar coordinates $\displaystyle r=\sqrt{x^2+y^2}$, $\displaystyle \theta=\tan^{-1}\left(\frac{y}{x}\right)$ andspecialize the resulting equation to the case when u does not not depend on the angular variable theta.

$\displaystyle \displaystyle u_{xx}=\left[\frac{\partial u}{\partial x}\frac{\partial x}{\partial r}+\frac{\partial u}{\partial x}\frac{\partial x}{\partial\theta}\right]^2$

$\displaystyle \displaystyle u_{yy}=\left[\frac{\partial u}{\partial y}\frac{\partial y}{\partial r}+\frac{\partial u}{\partial y}\frac{\partial y}{\partial\theta}\right]^2$

Changing to polar.

$\displaystyle \displaystyle u_{xx}=\left[u_r\frac{x}{r}-u_{\theta}\frac{y}{r}\right]^2=u_{rr}\frac{x^2}{r^2}-u_{r\theta}\frac{2xy}{r^3}+u_{\theta\theta}\frac{y ^2}{r^4}$

$\displaystyle \displaystyle u_{yy}=\left[u_r\frac{y}{r}+u_{\theta}\frac{x}{r}\right]^2=u_{rr}\frac{y^2}{r^2}+u_{r\theta}\frac{2xy}{r^3 }+u_{\theta\theta}\frac{x^2}{r^4}$

$\displaystyle \displaystyle u_{t}=k\left[u_{rr}\left(\frac{x^2+y^2}{r^2}\right)+u_{\theta\t heta}\left(\frac{x^2+y^2}{r^4}\right)\right]=k\left[u_{rr}+\frac{1}{r^2}u_{\theta\theta}\right]$

How do I make u not depend on theta?