In the book of Treves “Basic linear partial differential equations”, at page 169, the author writes:
“We are going the make the following assumption $$ (19.18) \qquad \hspace{6em} (grad_{\eta} P_{m})(y_0,\eta^0)\neq 0.$$ Then let $\theta$ be a vector ( in $\mathbb{R}^{n}$) not orthogonal to $ grad_{\eta} P_{m}(y_0,\eta^0)$. We may choose the coordinates in $\mathbb{R}^{n}$ in such a way $\theta$ becomes the unit vectors to the $y^N$-axis. This implies at once that $$\frac{\partial P_m}{\partial \eta_{_{N}}} \textit{ does not vanish in somme neighborhood } \mathcal{N} \textit{ of } (y_0, \eta^0).$$”
Could anyone help understand them? Thank you!