The expression on the left side is a material derivative.

Note that the Lagrangian rate of change is also known as the material derivative.

Note that the material derivative consists of two terms.

It turns out, however, that many physical concepts can be written concisely with the material derivative.

Objectivity rates are modified material derivatives that allows to have an objective time differentiation.

Note that the capital derivatives are the material derivatives.

The derivatives of p and L are material derivatives.

It is sometimes used as a synonym for the material derivative, , in fluid mechanics.

The material derivative is defined as the operator:

Instead, the material derivative is needed:

Let be the second material derivative of x. Then the d'Alembert-Euler condition is:

This equation expresses the material derivative of the field, that is, the derivative with respect to a coordinate system attached to the moving surface.

The material derivative can serve as a link between Eulerian and Lagrangian descriptions of continuum deformation.

When the flow is incompressible, does not change for any fluid parcel, and its material derivative vanishes: The continuity equation is reduced to:

That is, the path follows the fluid current described by the fluid's velocity field v. So, the material derivative of the scalar φ is:

The instantaneous position is a property of a particle, and its material derivative is the instantaneous velocity of the particle.

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

Applied to any physical quantity, the material derivative includes the rate of change at a point and the changes dues to advection as fluid is carried past the point.

We must then require that the material derivative of the density vanishes, and equivalently (for non-zero density) so must the divergence of the fluid velocity:

In the Lagrangian description, the material derivative of is simply the partial derivative with respect to time, and the position vector is held constant as it does not change with time.

A (dynamic) boundary condition in terms of only the potential Φ can be constructed by taking the material derivative of the dynamic boundary condition, and using the kinematic boundary condition:

The derivative of a field with respect to a fixed position in space is called the Eulerian derivative while the derivative following a moving parcel is called the advective or material derivative.

But also in the Eulerian frame of reference the notion of fluid parcels can be advantageous, for instance in defining the material derivative, streamlines, streaklines, and pathlines; or for determining the Stokes drift.

The material derivative of any property of a continuum, which may be a scalar, vector, or tensor, is the time rate of change of that property for a specific group of particles of the moving continuum body.

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).