I am trying to solve this problem:
Let $J^k (n,p)$ be the set of maps $\sigma:\mathbb{R}^n\rightarrow\mathbb{R}^p$, where $\sigma_i$ (the coordinate functions) is a polynomial of degree $\leq k$ with null constant term. We call such space as $k$-th jet bundle.
a) Let $\Sigma^i\subset J^1(n,1)$ be the set of $\sigma$ such that $\dim (\ker(\sigma))=i$. Show that $\Sigma^i$ is the empty set or is a submanifold. In the last case, calculate its dimension.
b) Given $f\in C^{\infty}(\mathbb{R}^n,\mathbb{R}^n)$, for each $p\in\mathbb{R}^n$ define the function $$j^1 f:\mathbb{R}^n\rightarrow J^1(n,1),$$ where $$j^1 f(p)(x_1,\dots, x_n)=\frac{\partial f(p)}{\partial x_1}x_1+\dots+\frac{\partial f(p)}{\partial x_n}x_n.$$
Show that $f$ is a Morse function if, and only if, $j^1 f \pitchfork\Sigma^n$ for each singular point $p$ of $f$.
So far what I did notice was that $J^1(n,1)$ can be identified with $\mathbb{R}^n$ and he admits only $\Sigma^{n}$ and $\Sigma^{n-1}$ not empty.