Consider a function $\vec u(y,t)$:$R^{n+1}→R^n$ and a coordinate transformation $\vec x(α,t)$:$R^{n+1}→R^n$ which also depends on time t.
Let u and x be related so that $\large \frac{∂x}{∂t}=u(x(α,t),t)$.
Let J denote the Jacobian of the coordinate transformation x, i.e. the entries of the Jacobian matrix are given by $J_{ij}=\frac{∂x_j}{∂α_i}$ and $J≡det[J_{ij}]$
Part 1:
Let $Jj≡∇x_j$ be the j-th column of the Jacobian matrix; show that $\large \frac{∂J_j}{∂_t}$=$∑_{n=1}^k \frac{∂u_j}{∂y_k}(\vec x)J_k$.
Part 2:
Show that $\large \frac{∂_J}{∂_t}=(∇⋅u)J.$
EDIT: I have shown Part 1 (was not hard but the setup of the answer was long and tedious, with lots of notation to keep track of, then simply use the multivariable chain rule) and am now working on Part 2.
Any hints or solutions are welcome for Part 2, especially if you can use the equation from Part 1 to solve Part 2.
Also, for Part 1, I assumed that the mixed partial derivatives were equal in order to use the relation $\large \frac{∂x}{∂t}=u(x(α,t),t)$, although nowhere in the problem statement mentions the smoothness of the functions in consideration. I do not see another way of solving Part 1, without making this assumption. So, I welcome any comments for Part 1, if you can shed some light into the problem statement.
Thanks,