What is the volume of the domain to which the cube $|x| < 1, |y| < 1, |z| < 1$ move to after 20 time units, when its motion is governed by the system
$\begin{cases}\dot x = \cos(x+y+z) \\ \dot y = -\sin(y+z) \\ \dot z = -\cos(x+y+z)+\sin(y+z)+2z\end{cases}$
Solution Let $\Omega(t)$ be the domain at time $t$. At $t=0$ this is the cube we were given, and the volume is $V(0) = \int_{\Omega(0)}dxdydz = 2^3 = 8$
Notice that the divergence of this system (the right side of the system) is $2$, and so:
$\dot V = \int_{\Omega(t)}\text{div}(F)dxdydz = 2\int_{\Omega(t)}dxdydz = 2V$
Hence $V(t) = V(0)e^{2t} = 8e^{2t}$ so our answer is $V(20) = 8e^{40}$
My issue is
I'm not sure why $\dot V = \int_{\Omega(t)}\text{div}(F)dxdydz$. I understood everything else.