Definition
Incompressible velocity field
A differentiable fluid velocity field with zero spatial divergence.
A differentiable fluid velocity field is incompressible when
throughout its domain. Thus each time slice is a divergence-free vector field.
Volume interpretation
For a smooth flow map, the Jacobian determinant satisfies
Since the determinant initially equals one, incompressibility preserves local volume while the flow map exists. The identity follows by differentiating the trajectory equation in and using the determinant derivative formula.
Density convention
The standard homogeneous incompressible model also assumes constant positive density. Divergence freedom is the kinematic constraint; constant density and the constitutive law are additional modeling assumptions.