Deformation gradient conservation law from Larangian to Eulerian formulation

In the following, I use the standard notation for (solid) mechanics and conservation laws, i.e. $ F$ the formation gradient, $ H$ the cofactor, $ v$ the velocity field and $ J$ the Jacobian. Moreover, $ X$ is for the initial configuration where functions and fields are expressed using a bar and $ x$ is for the mapped {current) configuration.

The rate of deformation gradient can be expressed as: $ \frac{\partial F}{\partial t} = \nabla \bar{v} $ .

Now I know how to derive the total (TLF) $ \frac{d}{dt}\int_V F \; dV = \int_{\partial V} \bar{v}\otimes N \; dA$ and updated (ULF) $ \int_{v(t)} J^{-1} \left.\frac{\partial F}{\partial t}\right\vert_{X} \; dv = \int_{v(t)} \nabla\cdot (v\otimes H) dv$ lagrangian formulations.

I would like to get, if it exists, the Eulerian formulation of the above, as well as the ALE and Total ALE formulation of it.

To do so, I tried to start from TLF, I switch to the current configuration $ v(t)$ using $ J$ and I use the Reynold’s transport theorem to get rid of the total time derivative on the integral and to get the convective term. I end up with:

$ \int_{v(t)} \frac{\partial J^{-1}F}{\partial t} + \nabla\cdot(J^{-1}F\otimes v) \; dv = \int_{v(t)} \nabla\cdot (v\otimes H^{-1}) \; dv$

Are the mathematics correct? And can I reach these formulations? May I say that I prefer integral forms..