I get confused in the following problem. Actually I don't know how to derive the derivative of the Jacobian. Could anybody help me?
Given a smooth vector field $\vec{b}$ on $R^{n}$, let $\vec{X}(s)=\vec{X}(s,x,t)$solves the ODE:
$\dot{\vec{X}}=\vec{b}(\vec{X})(s \in R)$,
$\vec{X}(t)=\vec{X}$
Define the Jacobian $J(s,x,t)=det D_{x}\vec{X}(s,x,t)$, derive Euler's formula:
$J_{s}=(div\vec{b}(\vec{X}))J$