Let $f:\mathbb{R}^k\to\mathbb{R}^n$ where $0<k\leq n$ are integers. Let $Q$ be a length $\binom{n}{k}$ vector consisting of the determinants for all $k\times k$ submatrices of the jacobian $J$ of $f$.
Is it possible to express $||Q||_2$ elegantly in terms of $J$ using matrix algebra?
E.g. if
$n=3$, $k=1$ then $||Q||_2$ is everywhere expressed as the norm of the gradient.
$n=3$, $k=2$ then $||Q||_2$ is everywhere expressed as the norm of a cross product.
$n=3$, $k=3$ then $||Q||_2$ is everywhere expressed as the absolute value of a determinant.
I don't like this diversity of expressions for similar concepts, but so far I could only eliminate it using words (first two lines).
Ok I see now $\ ||Q||_2 = \sqrt{\det(J^TJ)}$.
Edit: I just encountered the Cauchy–Binet formula and noted that the above equility is a special case.