Consider a function $f \in C^{\infty}(\mathbb{R}^n)$, $y \in Reg(f), M=f^{-1}(y)$. Prove that the canonical bundle of M is trivial.
I have an hint but I don't know how to use it: consider the open subset $M_i = \{p \in M \colon \frac {\partial f}{\partial x_i} \ne 0 \}$. Try to define $\omega_i \in \Omega^{n-1}(M)$ so that for all $p \in M_i$
$\omega_p =(-1)^i \frac {(dx_1 \wedge ... \wedge dx_{i-1}\wedge dx_{i+1}\wedge ... \wedge dx_n)_p} {\frac {\partial f}{\partial x_i}}$.