Scalar metric Jacobian and weak-gradient identification

Work in ZFC. For a real-valued Lipschitz function $u$ on a smooth Riemannian three-region $(R,k)$, at every differentiability point the scalar coarea Jacobian of $du:T_{x}R\to \mathbb{R}$ is $J_{1}u=\mathrm{\mid }du{\mathrm{\mid }}_k=\mathrm{\mid }{\mathrm{\nabla }}_{k}u\mathrm{\mid }$. At points with $du=0$ the value is zero. This is the one-dimensional-target (rank at most one) Jacobian, not an endpoint-map two-dimensional area Jacobian.

Precisely, for a linear functional $\mathrm{\ell }:T_{x}R\to \mathbb{R}$, let ${\mathrm{\ell }}^{\mathrm{♯}}$ be the unique vector satisfying $k_x({\mathrm{\ell }}^{\mathrm{♯}},v)=\mathrm{\ell }(v)$ for every $v\in T_{x}R$, and put $\mathrm{\mid }\mathrm{\ell }{\mathrm{\mid }}_k=\sqrt{k_x({\mathrm{\ell }}^{\mathrm{♯}},{\mathrm{\ell }}^{\mathrm{♯}})}$. Its adjoint ${\mathrm{\ell }}^{\ast }:\mathbb{R}\to T_{x}R$ is characterized by $k_x({\mathrm{\ell }}^{\ast }t,v)=t\mathrm{\ell }(v)$, with the usual Euclidean inner product on $\mathbb{R}$. Define the scalar metric coarea Jacobian by $J_1(\mathrm{\ell }):=\sqrt{\det (\mathrm{\ell }{\mathrm{\ell }}^{\ast })}$, where the determinant is on the one-dimensional target, and set $J_{1}u(x)=J_1(d{u}_x)$ at differentiability points. There ${\mathrm{\nabla }}_{k}u=(d{u}_x{)}^{\mathrm{♯}}$; almost everywhere this agrees with the metric weak gradient.