= Jacobian evolution of a smooth flow
{c}
{title2=$\partial_tJ[\varphi_t]=(\nabla\cdot F)(\varphi_t)J[\varphi_t]$}
For a smooth <flow map> $\varphi_t$ generated by a <vector field> $F$, the <chain rule> and $\det(I+tDF)=1+t\operatorname{tr}(DF)+o(t)$ give $\partial_tJ[\varphi_t](x)=(\nabla\cdot F)(\varphi_t(x))J[\varphi_t](x)$. With initial value one, its solution is the exponential of the time integral of this <divergence>. Thus a divergence-free generator preserves Euclidean volume and orientation. This connects the <Jacobian determinant> with a <volume-preserving vector field>.
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