Let the standard coordinate system of $\;\mathbb R^2\;\;,\;\{x,y\}\;$ and a rectangle $\;\mathcal R\;$. Take $\;x^* \;$ on the boundary of $\;\mathcal R\;$ and consider there the positive oriented orthonormal basis $\;\{τ,v\}\;$ where $\;v\;$ is the outward unit normal vector and $\;τ\;$ the tangent one.
Is it true to claim that:
$\;\frac{\partial f}{\partial y}=\frac{\partial f}{\partial v}\;$ for some non zero and non constant function $\;f\;$ and if yes,why?
It seems valid to me, although I'm missing the key in order to prove it. The change of coordinates and how this affects calculations, always confuses me a lot. Any help would be valuable!
Thanks in advance