convective derivative

The convective derivative takes into account changes due to time dependence and motion through space along vector field.

Noting that what remains on the left side of the equation is the convective derivative:

For this reason the convective derivative is also known as the particle derivative.

Being small in comparison, diffusion and the "convective derivative" (second term on the left) can be left out.

The portion of the material derivative represented by the spatial derivatives is called the convective derivative.

The material derivative is also known as the substantial derivative, or comoving derivative, or convective derivative.

Since a streamline is a curve that is tangent to the velocity vector of the flow, the left-handed side of the above equation, the convective derivative of velocity, can be described as follows:

The Lagrangian and Eulerian specifications of the kinematics and dynamics of the flow field are related by the substantial derivative (also called the Lagrangian derivative, convective derivative, material derivative, or particle derivative).

That is, viewing the motion through phase space as a 'fluid flow' of system points, the theorem that the convective derivative of the density, , is zero follows from the equation of continuity by noting that the 'velocity field' in phase space has zero divergence (which follows from Hamilton's relations).