let $\phi :\mathbb{R^+\times R^3} \to \mathbb{R^3} $ defined as follows
for all $(t,X)\in\mathbb{R^+\times R^3}$ we have: $\phi(t,X)=(\phi_1(t,X),\phi_2(t,X),\phi_3(t,X))$. And we define the Jacobian Matrix of $\phi(t,.):$
$$ J_t(X)= \left[ {\begin{array}{ccc} \frac{\partial \phi_1}{\partial {x_1}} & \frac{\partial \phi_1}{\partial {x_2}} & \frac{\partial \phi_1}{\partial {x_3}} \\ \frac{\partial \phi_2}{\partial {x_1}} & \frac{\partial \phi_2}{\partial {x_2}} & \frac{\partial \phi_2}{\partial {x_3}}\\ \frac{\partial \phi_3}{\partial {x_1}} & \frac{\partial \phi_3}{\partial {x_2}} & \frac{\partial \phi_3}{\partial {x_3}}\\ \end{array} } \right] $$ And I want to prove that : $\partial_tJ_t(X)=div[\frac{d}{dt}\phi(t,X)]J(t,X)$ My idea: $$ \partial_tJ_t(X)= \left[ {\begin{array}{ccc} \frac{\partial}{\partial t}\frac{\partial \phi_1}{\partial {x_1}} & \frac{\partial}{\partial t}\frac{\partial \phi_1}{\partial {x_2}} & \frac{\partial}{\partial t}\frac{\partial \phi_1}{\partial {x_3}} \\ \frac{\partial}{\partial t} \frac{\partial \phi_2}{\partial {x_1}} & \frac{\partial}{\partial t}\frac{\partial \phi_2}{\partial {x_2}} & \frac{\partial}{\partial t}\frac{\partial \phi_2}{\partial {x_3}}\\ \frac{\partial}{\partial t} \frac{\partial \phi_3}{\partial {x_1}} & \frac{\partial}{\partial t}\frac{\partial \phi_3}{\partial {x_2}} & \frac{\partial}{\partial t}\frac{\partial \phi_3}{\partial {x_3}}\\ \end{array} } \right] $$ But it did not work.