Assuming we have a function $u(x,t)$ and using the change of variables $\xi=x-t$ and $\eta=x+t$ we define a new function $v(\xi,\eta)=u(x,t)$.

I am attempting to take compute the partial derivatives $\frac{\partial^2u}{\partial t^2}$ and $\frac{\partial^2u}{\partial x^2}$. For the first order partial derivatives I have found that: \begin{equation} \frac{\partial u}{\partial t}=\frac{\partial v}{\partial \xi}\frac{\partial \xi}{\partial t}+\frac{\partial v}{\partial \eta}\frac{\partial \eta}{\partial t}=-\frac{\partial v}{\partial \xi}+\frac{\partial}{\partial \eta} \end{equation} Similarly, for differentiation w.r.t $x$ I have: \begin{equation} \frac{\partial u}{\partial x}=\frac{\partial v}{\partial \xi}+\frac{\partial v}{\partial \eta} \end{equation} Now for the second order derivatives I have used: \begin{equation} \frac{\partial^2 u}{\partial^2 t}=\frac{\partial}{\partial t}\left( -\frac{\partial v}{\partial \xi}+\frac{\partial}{\partial \eta} \right)=-\left( \frac{\partial^2 v}{\partial \xi^2}\frac{\partial \xi}{\partial t} +\frac{\partial^2 v}{\partial \xi \partial\eta}\frac{\partial \eta}{\partial t}\right)+\left( \frac{\partial^2 v}{\partial \eta^2}\frac{\partial \eta}{\partial t} +\frac{\partial^2 v}{\partial \eta \partial\xi}\frac{\partial \xi}{\partial t}\right)=\frac{\partial^2v}{\partial \xi^2}-2\frac{\partial^2 v}{\partial \eta\partial\xi}+\frac{\partial^2 v}{\partial \eta^2} \end{equation} Is my attempt correct or I have missed something?


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