Flow map

For an autonomous ordinary differential equation, the flow map ${\phi }_t$ sends each initial state to its state after time $t$. Uniqueness gives ${\phi }_{t+s}={\phi }_t\circ {\phi }_s$ and ${\phi }_{-t}={\phi }_t^{-1}$ whenever both sides exist.

For a constant real square matrix $A$, the autonomous differential equation $\dot{x}=Ax$ has the linear flow ${\phi }_t(x)=e^{tA}x$, where $e^{tA}$ is the matrix exponential. The identity $e^{(t+s)A}=e^{tA}{e}^{sA}$ gives the flow map composition law, and $e^{-tA}$ reverses the evolution. Its determinant is $e^{t\mathrm{tr}A}$, so the matrix trace gives the instantaneous fractional rate of volume change, as illustrated in trace and local volume change. A zero trace preserves volume even when the flow stretches one direction and contracts another.

For a smooth flow map ${\phi }_t$ generated by a vector field $F$, the chain rule and $\det (I+tDF)=1+t\mathrm{tr}(DF)+o(t)$ give ${\partial }_{t}J[{\phi }_t](x)=(\nabla \cdot F)({\phi }_t(x))J[{\phi }_t](x)$. 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.