Using the chain rule to find second order partial derivatives

Question

Given the application $u(r,\phi):=v(r\cos\phi, r\sin \phi)$ I need to find by the chain rule an expression for $\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2}$ in terms of second partial derivatives of $u.$ I found first the Jacobian matrix based on the transformation function $(r,\phi)\rightarrow (x,y)=(r\cos \phi, r \sin \phi).$ Since this application is not a function, I do not know if one has to go through the Hessian matrix to find the second partial derivatives. Can you provide me some support or a solution proposal ? Thanks.

Answer 1

Hint: Since you want an expression of the second order partial derivatives of $u$, begin by finding what they are in terms of the p.d. of $v$ and the derivatives of $x$ and $y$.

Reason: the first and second order partial derivatives of $x, y$ with respect to $r, \phi$ are much easier to find than the inverse, and trigonometric identities may be useable.$$\dfrac{\partial x}{\partial r}=\cos\phi~, \dfrac{\partial x}{\partial \phi}=-r\sin\phi~, \dfrac{\partial^2 x}{\partial r~^2}=0~,\textit{et cetera}\ldots$$