If z=f(x,y), where x=rcosθ and y=rsinθ, show that (∂^2 z/∂y^2)=(∂^2 z/∂r^2)=(1/r^2)(∂^2 z/∂θ^2)+(1/r)(∂^2 z/∂r)

Answer 1

Where do you get stuck? This is similar to ones you've posted, and I'm sure google can provide discussions on second derivatives of polar coordinates. In fact, I already sent you one reference, dealing with the Laplacian. Post some work, and maybe in the process you can clear things up, or we can help iron out the difficulties. This is a bit tedious, but it's nothing more than applying the rules you have learned so far.