# Convective derivative vs total derivative

I was wondering what is the difference between the convective/material derivative and the total derivative. We were introduced to the notion of material derivative

$$\frac{D\vec{u}}{Dt}=\frac{\partial \vec{u}}{\partial t}+(\vec u.\vec{\triangledown})\vec u$$

in fluid mechanics but I can not see how this is any different to the time derivative of a vector field

$$\vec{u}({\vec{x(t)}},t)$$ and just applying the chain rule?

If there is no difference, why introduce new notation of $$\frac{ D\vec{u}}{D t}$$ instead of just using the standard notation $$\frac{d\vec{u}}{dt}$$?

• Different authors use different notation. – Qmechanic Apr 19 at 12:27

## 1 Answer

Take a look at the definitions below

Total derivative: $$\frac{d}{dt} = \frac{\partial}{\partial t} + \frac{dx_i}{dt}\frac{\partial}{\partial x_i}$$

Material derivative: $$\frac{D}{Dt} = \frac{\partial}{\partial t} + u_i\frac{\partial}{\partial x_i}$$

Would you agree they are the same if $$u_i=\frac{dx_i}{dt}$$?

Why people use inconsistent notation for both? I do not know.

In my opinion the total derivative is used most often in mathematics whereas the material derivative is used most often in physics. In physics, $$\frac{dx_i}{dt}$$ has a clear physical interpretation as the instantaneous velocity. In mathematics, there is not necessarily a physical interpretation and the variable notation may be arbitrary.

The next question is, do you understand what the physical interpretation of the material derivative is?

• I believe physically it describes how a quantity (scalar or vector) of an element changes as the element moves in time and space? – Dan Apr 25 at 11:00