Partial Derivatives with respect to independent variables

Question

Could I please get help with the following question?

Let $ = + $ and $ = − $ Express $\frac{^2}{^2}$ and $\frac{^2}{^2}$ in terms of partial derivatives of $$ with respect to the independent variables $$ and $$.

Answer 1

By the chain rule we have

$$\frac{\partial f}{\partial x}=\frac{\partial f}{\partial \xi}\frac{\partial \xi}{\partial x}+\frac{\partial f}{\partial \eta}\frac{\partial \eta}{\partial x}=\frac{\partial f}{\partial \xi}+\frac{\partial f}{\partial \eta}$$ $$\frac{\partial^2 f}{\partial x^2}=\frac{\partial}{\partial x}\left(\frac{\partial f}{\partial \xi}+\frac{\partial f}{\partial \eta}\right)=\frac{\partial}{\partial \xi}\left(\frac{\partial f}{\partial \xi}+\frac{\partial f}{\partial \eta}\right)+\frac{\partial}{\partial \eta}\left(\frac{\partial f}{\partial \xi}+\frac{\partial f}{\partial \eta}\right)$$ $$=\frac{\partial^2f}{\partial\xi^2}+\frac{\partial^2f}{\partial\xi\partial\eta}+\frac{\partial^2f}{\partial\eta\partial\xi}+\frac{\partial^2f}{\partial\eta^2}$$

etc.

Answer 2

Use the fact that $\displaystyle\frac{\partial f}{\partial x}=\frac{\partial f}{\partial \zeta}\frac{\partial \zeta}{\partial x}+\frac{\partial f}{\partial \eta}\frac{\partial \eta}{\partial x}$