Euler's volume identity ($J_t=J\nabla \cdot \mathbf{u}$)

For a smooth, locally invertible material map $\mathbf{x}=\mathbf{X}(\mathbf{a},t)$ with ${\mathbf{X}}_t=\mathbf{u}(\mathbf{X},t)$ and positive Jacobian determinant $J=\det (\partial \mathbf{X}/\partial \mathbf{a})$, Euler's volume identity states that $J_t=J\nabla \cdot \mathbf{u}$. Indeed, $F=\partial \mathbf{X}/\partial \mathbf{a}$ obeys $F_t=(\nabla \mathbf{u})F$, so the determinant derivative is $J_t=J\mathrm{tr}(F^{-1}{F}_t)=J\mathrm{tr}(\nabla \mathbf{u})$. This form of the Jacobian evolution of a smooth flow applies also to explicitly time-dependent velocity fields. It gives the material-volume Reynolds transport theorem by change of variables to the fixed particle labels. It is distinct from the complex-exponential Euler's formula.