Consider the following step of p. 499 from Lax's proof of the change of variables theorem:
Lemma
Let $\phi \in C^{2}(\mathbb{R}^n; \mathbb{R}^2)$ and denote by $D\phi_j$ the gradient of the j-th component of $\phi$. Let $D= (\partial_{x_1}, \dots, \partial x_n )^T$.
Then
$$ \det((D,D\phi_2, \dots , D\phi_n )) = \sum_{2 \le k \le n} \det((D,D\phi_2, \dots, D\phi_n ))_k $$
Where the subscript k means that the differential operator D in the first column acts only on the k-th column. Moreover each determinant in the sum is zero.
I mean, I wrote it down in dimension 3 but a rigorous proof is another thing!
Any suggestion?