Let $M \subseteq \mathbb{R}^n$ be an open set, and let $u \in C([0,T]; W^{1,\infty}(M; \mathbb{R}^n))$ be a velocity field. Let $\Phi: [0,T] \times M \to M$ denote the flow map associated to $u$, defined as the unique solution of \begin{align*} \frac{d}{dt}\Phi(t,x) &= u(t, \Phi(t,x)), \quad \Phi(0,x) = x, \end{align*} for each $x \in M$. Define the spatial Jacobian matrix $\hat{J}(t,x) \in \mathbb{R}^{n \times n}$ by $\hat{J}_{ij}(t,x) = \frac{\partial \Phi_i}{\partial x_j}(t,x)$, and the Jacobian determinant $\mathcal{J}: [0,T] \times M \to \mathbb{R}$ by $\mathcal{J}(t,x) = \det(\hat{J}(t,x))$. Then $\mathcal{J}$ satisfies \begin{align*} \partial_t \mathcal{J}(t,x) &= \mathcal{J}(t,x)\,(\nabla \cdot u)(t, \Phi(t,x)), \end{align*} pointwise in $(t,x) \in [0,T] \times M$.