Is there a name for $(\det JJ^\top)^{1/2}$ where $J$ is a non-square Jacobian matrix?

Question

Background: Let $f: \mathbb R^n \to \mathbb R^m$ be a differentiable vector-valued function, with components $f_i: \mathbb R^n \to \mathbb R, \textrm{ for }1\le i\le m$. Then the Jacobian matrix is the $m \times n$ matrix $$\textbf{J} = \begin{bmatrix} \dfrac{\partial f_1}{\partial x_1} & \cdots & \dfrac{\partial f_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial f_m}{\partial x_1} & \cdots & \dfrac{\partial f_m}{\partial x_n} \end{bmatrix}$$

As is well-known, in the case in which $n = m$ the Jacobian matrix is square, and its determinant, the Jacobian determinant, is the scalar $J = \det \textbf{J}$. The Jacobian determinant has many uses, including when using a substitution during integration.

But what if $m \ne n$? In this case, $\textbf{J}$ is not square, so $\det \textbf{J}$ is not defined. However, in this case, the quantity $\left(\det \textbf{J} \textbf{J}^\top \right)^{1/2}$ is still meaningful, and generalizes the idea (and some of the uses) of the Jacobian determinant:

1. For example, consider a function $\vec r: \mathbb R^2 \to \mathbb R^3$, which we can regard as a parametrization of a surface in $\mathbb R^3$. If we write $\vec r(u,v) = \langle x(u,v), y(u,v), z(u,v) \rangle$, then the Jacobian matrix is $\textbf{J} = \begin{bmatrix} x_u & y_u & z_u \\ x_v & y_v & z_v \end{bmatrix}$; in this situation it can be shown that $\left(\det \textbf{J} \textbf{J}^\top \right)^{1/2} = \| \vec r_u \times \vec r_v \|$. We recognize the quantity that relates area in the surface to area in the domain: $dS = \| \vec r_u \times \vec r_v \| \, dA$.
2. On the other hand consider a function $\vec r: \mathbb R^1 \to \mathbb R^n$, which we can interpret as a parametrization of a path in $\mathbb R^n$. If we write $\vec r(t) = \langle x_1(t), x_2(t), \dots, x_n(t) \rangle$ then in this case $\left(\det \textbf{J} \textbf{J}^\top \right)^{1/2} = \left| \frac{d\vec r}{dt} \right|$. Once again we recognize the quantity that relates arc length in the path to length in the domain: $ds = \left| \frac{d\vec r}{dt} \right| \, dt$.
3. Finally we note that if $n = m$ then $\left(\det \textbf{J} \textbf{J}^\top \right)^{1/2}$ is identical to the "usual" Jacobian determinant $J$.

My question:

Does the quantity $\left(\det \textbf{J} \textbf{J}^\top \right)^{1/2}$ have a standard name in the case of a non-square Jacobian matrix $\textbf{J}$? The phrase "Jacobian determinant" seems, as far as I can tell, to only be used in the context of square matrices, despite the fact that the quantity generalizes quite naturally to the non-square case, and encompasses several "special formulas" in a single neat form. But I am unaware of any established terminology for this quantity. Is there one?

Answer 1

Im not sure if you're already aware of this, but $J^TJ$ is actually a matrix representation of the metric tensor $g$ in some coordinates. Indeed, recall that the column vectors in the Jacobian matrix represent the new basis vectors in terms of the old ones, hence $(J^TJ)_{ij}=e_i.e_j$.

That is, the components of $J^TJ$ exactly represent the dot product of the new basis vectors, which is exactly what the components of the metric tensor represent.

Now when $n\ne m$ like in your case then $J^TJ$ represents the induced metric, and hence its determinant is just the determinant of the induced metric.