Transversality and the hessian of critical ponint of a function I am trying to do the following exercise:
Let $f\in C(\mathbb{R^n,\mathbb{R}})$. Consider the gradient vector field $\nabla f=(\frac{\partial f}{\partial x_1},...,\frac{\partial f}{\partial x_n})$. Let $\Gamma_{\nabla f}\subset $ be it's graph, and let $\mathbb{R}^n $ be embedded in $\mathbb{R}^{2n}$ by $x\rightarrow (x,0)$. Show that $\Gamma_{\nabla f} \pitchfork \mathbb{R}^n$ if and only if and the critical points of $f$ are non-degenarate.
Now first for we need to figure out what is the tangent space of $\Gamma_{\nabla f}$. Well this will be a submanifold in $\mathbb{R}^{2n}$ by considering the function $G(x,y):\mathbb{R}^{2n}\rightarrow \mathbb{R}^n $ , $(x,y)\rightarrow y-\nabla_f$ and checking that $0$ is a regular value for this map. Now we know from the regular value theorem that $T_p (\Gamma_{\nabla f})=\ker d_pG $. Now the $\ker$ of this map will be the vectors in $\mathbb{R}^{2n}, (v_1,...,v_{2n}) $ such that $H_f(p)(v_1,...,v_n)=(v_{n+1},...,v_{2n})$.
Now I still need to finish the problem but is this the right way to be thinking about this ? Thanks in advance.
 A: Consider $\Gamma_{\nabla f} : \mathbb{R}^{n}\rightarrow \mathbb{R}^{2n}$, defined by
$$\Gamma_{\nabla f} (x) = \big(x_{1},...,x_{n},\frac{\partial f}{\partial x_{1}}(x),\frac{\partial f}{\partial x_{2}}(x),...,\frac{\partial f}{\partial x_{n}}(x)\big)$$
Notice that when $p\in \mathbb{R}^{n}$ is a critical point for $f$, $\Gamma_{\nabla f}$ intersects the plane $\mathbb{R}^{n}\times \{0\}$ at $(x,0)$. Then $\Gamma_{\nabla f}\pitchfork(\mathbb{R}^{n}\times \{0\})$ if and only if $0$ is a regular value for the map $g\circ \Gamma_{\nabla f}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ where $g:\mathbb{R}^{2n}\rightarrow \mathbb{R}^{n}$ given by $g(x,y)=y$. The only way a value of map from n dimensional space to n dimensional space is a regular value is if the derivative map is invertible. Computing the derivative we get the Hessian matrix at $x$:
$$
\begin{pmatrix}
\frac{\partial ^{2} f}{\partial x_{i} \partial x_{j}}(x)
\end{pmatrix}.
$$
The Hessian invertible at a critical point is the definition of a critical point being nondegenerate.
