On the integral of Frobenius norm of Jacobian Let $g : S \to \mathbb{R}^d$ be injective and smooth where $S \subseteq \mathbb{R}^k$ is a $k$-dim regular smooth compact manifold with $d \geq k$.
I'de considering the term
$$
A = \int_S \lVert J_g(x) \rVert_F \, dx
$$
where $J_g$ is Jacobian matrix of $g$ and $\lVert \cdot \rVert_F$ is Frobenius norm.
If $S$ is an one-dimensional interval on $\mathbb{R}$, then $A$ is a the length of the curve given by $g(S)$.
I have following questions:

*

*What would be a physical meaning of $A$?

*Does increasing the $k$-dimensional area of $g(S)$ increases $A$, and vice versa?

*If 2 is true, how to simply prove?

Hopefully my question is not so stupid one.

$\|J_g(x)\|^2$ is often called the energy density at $x$ of the map $g$. Here, the constant map is viewed as having the least density, so the energy can be viewed as measuring the energy due to stretching $S$. The integral of the energy density is therefore called the total energy. Given some additional constraints on the map $g$ (so, for example, it isn't allowed to be the constant map), it is natural to ask when $g$ is energy-minimizing. Energy-minimizing maps are known as harmonic maps.
You can also treat $\|J_g\|^p$, including $p = 1$, as a type of energy density and study so-called $p$-harmonic maps. This is done less, simply because $p=2$ is the case that arises more naturally in physics and is also much easier to study mathematically.
The area of $g(S)$ is given by the integral of $\det J_g$. Since it's possible to increase $\det J_g$ but not $\|J_g\|$ and vice versa, there's no reason why the change in area should force a change in the energy or vice versa.
