We have been given $x=r\cos t$ and $y=r\sin t$ and asked to prove that $\partial^2t/\partial x^2+\partial^2t/\partial y^2=0$.
Assuming $r$ and $t$ to be dependent variables and $x$ and $y$ to be independent variables we proceed and obtain a relation containing $x, y$ and $t$ i.e $t=\arctan(y/x)$. Now we partially differentiate this w.r.t $x$ twice to obtain $\partial^2t/\partial x^2$. Similarly $\partial^2t/\partial y^2$ is also found out and then both the terms are added to reach the desired result.
Well and good till here.
But, a problem arises if we perform implicit partial differentiation on $\tan t=y/x$ to find $\partial^2t/\partial x^2$ and $\partial^2t/\partial y^2$. If this how we proceed then the two terms never add to be zero but instead give some non zero quantity. I get this:
$$\frac{\partial^2t}{\partial x^2}+\frac{\partial^2t}{\partial y^2}=-2\frac{\sin^3(t)}{r^2\cos(t)}+\frac{\tan(t)}{r^2}-2\frac{\sin(t)}{r}$$
My prof had told to separate out the dependent variable explicitly and then proceed with partial differentiation in case of polar co-ordinates but didn't actually tell us why. I thought he was saying so just because venturing otherwise would make stuff more cumbersome but after attempting this question it looks like I am missing a bigger picture.
P.S:- Please don't close the question just because formatting isn't proper. I don't know how to use latex and I am very sorry for that.
