= Euler's volume identity
{c}
{title2=$J_t=J\nabla\cdot\mathbf u$}
= Euler's identity for a material volume
{c}
{synonym}
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 $J_t=J\operatorname{tr}(F^{-1}F_t)=J\operatorname{tr}(\nabla\mathbf u)$. 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>.
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