As introduction to solving the one dimensional wave function our lecturer introduced us to transforming coordinates in a different way than we used to.
He elaborated the usage of the differential of a function for coordinate transformation by an example:
He defines the following coordinate transfomration: \begin{equation}\tag{1} \eta = x+y \qquad \xi = x-y \end{equation} \begin{equation}\tag{2} df = \left(\frac{\partial f}{\partial \eta}\right)_{\xi} d\eta + \left(\frac{\partial f}{\partial \xi}\right)_{\eta} d\xi \end{equation}
He then states that (2) which is apparently just the definition of the differential of a function equals to the following: \begin{equation}\tag{3} df = \left(\frac{\partial f}{\partial \eta}\right)_{\xi} d\eta \end{equation} He justifies that because $d\xi=dx-dy=0$
We discussed that justification for an hour now and everyone got an argument that he thinks is valid but in each and every argument we found a little flaw.
We hope someone can give us the right argument for this justification.
Unless your x and y mean something special, that final statement is not valid. From discussion in the comments, all I can think of is causality, but without boundary conditions on which it could be based, it can't be justified and in no way is true in general. Go back to your levturer. The onus is on them to explain things clearly.