I'm reading the following proposition of Introducion to PDE's of G. Folland but I don't understand the following part:
For each $x\in S$ its Jacobian matrix at $(x,0)$ is nonsingular since $\nu$ is normal to $S$.
My attempt to deduce this:
$$F(x,t):\mathbb{R}^n\times\mathbb{R}\to V\subseteq\mathbb{R}^n$$ where $$ F(x,t)= \left(x_1+\frac{t}{|\nabla \phi(x)|}\frac{\partial \phi}{\partial x_1}(x),\ldots,x_n+\frac{t}{|\nabla \phi(x)|}\frac{\partial \phi}{\partial x_n}(x)\right) $$ The Jacobian matrix is $$ J_f(x,t)\begin{pmatrix} 1+\frac{t}{|\nabla \phi(x)|}\frac{\partial^2 \phi}{\partial x_1^2}(x) & \cdots & 1+\frac{t}{|\nabla \phi(x)|}\frac{\partial^2 \phi}{\partial x_1\partial x_n}(x) & \frac{1}{|\nabla \phi(x)|}\frac{\partial \phi}{\partial x_1}(x) \\ \vdots & \ddots & \vdots & \vdots \\ 1+\frac{t}{|\nabla \phi(x)|}\frac{\partial^2 \phi}{\partial x_1\partial x_n}(x) & \cdots & 1+\frac{t}{|\nabla \phi(x)|}\frac{\partial^2 \phi}{\partial x_n^2}(x) & \frac{1}{|\nabla \phi(x)|}\frac{\partial \phi}{\partial x_n}(x) \end{pmatrix} $$ and therefore $$ J_f(x,0)\begin{pmatrix} 1 & \cdots & 0 & \frac{1}{|\nabla \phi(x)|}\frac{\partial \phi}{\partial x_1}(x) \\ \vdots & \ddots & \vdots & \vdots \\ 0 & \cdots & 1& \frac{1}{|\nabla \phi(x)|}\frac{\partial \phi}{\partial x_n}(x) \end{pmatrix} $$
I would appreciate any suggestion.