I have the following question:
Let $u(x, y)$ be a function with continuous second order partial derivatives. Use the chain rule to transform the expression:
$$ x^2\frac{\partial^2u}{\partial y^2}-xy\frac{\partial^2u}{\partial y\partial x}+x\frac{\partial u}{\partial y} $$
into polar coordinates.
Best Answer
This is what I did, but I'm not sure if this is right.
First I find the second-order partial derivatives, by using the chain rule:
Now using the fact that $x=r \cos(\theta)$ and $y=r \sin(\theta)$, I find $\frac{\partial u}{\partial x}$ and $\frac{\partial u}{\partial y}$ in terms of $r$ and $\theta$:
From here it's just a simple case of plugging in terms into the expression, which I am too lazy to do. Can anyone confirm that this is indeed correct?