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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.

Let $V(t)$ be a smoothly moving material volume, carried by a smooth velocity field $\mathbf{u}$, and let $f$ be continuously differentiable. The material-volume form of the Reynolds transport theorem statesEquivalently, the divergence theorem giveswhere $\mathbf{n}$ is the outward unit normal vector. The second integral measures the flux carried across the instantaneous boundary by its motion.

For the proof, use the particle map $\mathbf{x}=\mathbf{X}(\mathbf{a},t)$ from $V(0)$, with positive Jacobian determinant $J=\det (\partial \mathbf{X}/\partial \mathbf{a})$. Euler's volume identity isIt is the Jacobian evolution of a smooth flow: if $F=\partial \mathbf{X}/\partial \mathbf{a}$, differentiation gives $F_t=(\nabla \mathbf{u})F$, and differentiation of the determinant gives $J_t=J\mathrm{tr}(F^{-1}{F}_t)=J\nabla \cdot \mathbf{u}$. The change of variables formula now moves the integral to the fixed reference volume:Finally, $Df/Dt+f\nabla \cdot \mathbf{u}={\partial }_{t}f+\nabla \cdot (f\mathbf{u})$, giving the equivalent boundary form.