In ${\mathbb R}^n$, let $F$ be a smooth one-to-one mapping of $\Omega$ onto some open set \Omega', where $\Omega\subset{\mathbb R}^n$ is open. Set $y=F(x)$. Assume that the Jacobian matrix $J_x=[(\partial y_i/\partial x_j)(x)]$ is nonsingular for $x\in\Omega$. We have $\frac{\partial}{\partial x_j}=\sum\frac{\partial y_i}{\partial x_j}\frac{\partial}{\partial y_i}.$
Here are my questions:
Is there a neat way to calculate $\frac{\partial^2}{\partial x_j\partial x_k}?$ After several steps trial, I am completely confused. More generally, what is $\partial_x^{\alpha}$ in terms of the $y$ coordinate system? Here $\partial_x^{\alpha}:=\frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1}\cdots\partial x_n^{\alpha_n}}.$