Let $x,y$ be independent variables, and we transform into $\xi = \xi(x,y), \eta = \eta(x,y)$. By the chain rule, we clearly have:$$ \frac{\partial u}{\partial x}=\frac{\partial u}{\partial \xi} \frac{\partial \xi}{\partial x}+\frac{\partial u}{\partial \eta} \frac{\partial \eta}{\partial x}, \quad \frac{\partial u}{\partial y}=\frac{\partial u}{\partial \xi} \frac{\partial \xi}{\partial y}+\frac{\partial u}{\partial \eta} \frac{\partial \eta}{\partial y}.$$Here is the identity that I'm trying to derive:$$ \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial \xi^{2}}\left(\frac{\partial \xi}{\partial x}\right)^{2}+2 \frac{\partial^{2} u}{\partial \xi \partial \eta} \frac{\partial \xi}{\partial x} \frac{\partial \eta}{\partial x}+\frac{\partial^{2} u}{\partial \eta^{2}}\left(\frac{\partial \eta}{\partial x}\right)^{2}+\frac{\partial u}{\partial \xi} \frac{\partial^{2} \xi}{\partial x^{2}}+\frac{\partial u}{\partial \eta} \frac{\partial^{2} \eta}{\partial x^{2}}$$We have $\frac{\partial (\cdot)}{\partial x}=\frac{\partial (\cdot)}{\partial \xi} \frac{\partial \xi}{\partial x}+\frac{\partial (\cdot)}{\partial \eta} \frac{\partial \eta}{\partial x}$, thus$$\frac{\partial^{2} (\cdot)}{\partial x^{2}} = \left(\frac{\partial (\cdot)}{\partial \xi} \frac{\partial \xi}{\partial x}+\frac{\partial (\cdot)}{\partial \eta} \frac{\partial \eta}{\partial x} \right)\left(\frac{\partial (\cdot)}{\partial \xi} \frac{\partial \xi}{\partial x}+\frac{\partial (\cdot)}{\partial \eta} \frac{\partial \eta}{\partial x} \right).$$I multiplied this out, and made sure that the differential operators in the left bracket operated on the terms in the right bracket, and eventually I simplified to:$$\frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial \xi^{2}}\left(\frac{\partial \xi}{\partial x}\right)^{2}+2 \frac{\partial^{2} u}{\partial \xi \partial \eta} \frac{\partial \xi}{\partial x} \frac{\partial \eta}{\partial x}+\frac{\partial^{2} u}{\partial \eta^{2}}\left(\frac{\partial \eta}{\partial x}\right)^{2},$$where the $+\frac{\partial u}{\partial \xi} \frac{\partial^{2} \xi}{\partial x^{2}}+\frac{\partial u}{\partial \eta} \frac{\partial^{2} \eta}{\partial x^{2}}$ is nowhere to be seen. I've seen the proper way to derive these equations in this post: PDE standard form chain rule derivations, and I wanted to know where I had gone wrong in my attempt. Could anyone help? Thanks.
My attempt: Note $\partial_x (\partial_x)=(\xi_x \partial_\xi +\eta_x \partial_\eta)(\xi_x \partial_\xi +\eta_x \partial_\eta)$, so$$ \partial_x (\partial_x)= \xi_x \left(\xi_{x\xi} \partial_\xi + \xi_{x} \partial_{\xi\xi} + \eta_{x\xi} \partial_\eta + \eta_x \partial_{\eta\xi} \right) + \eta_x\left(\xi_{x\eta}\partial_\xi +\xi_x \partial_{\xi\eta} + \eta_{x\eta}\partial_\eta+ \eta_x\partial_{\eta\eta} \right) $$Now, $\xi_{x\xi} = \xi_{\xi x} = \partial_x(\frac{\partial \xi}{\partial\xi}) = \partial_x (1) = 0$. Similarly, $\xi_{x\eta} = \xi_{\eta x} = 0$ as $\frac{\partial \eta}{\partial \xi} = 0$. Proceeding in this way, things simplify to my incorrect attempt.
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