# The Material Time Derivative — Lesson 5

The video starts with a discussion showing that you can rewrite the Lagrangian and Eulerian descriptions in terms of each other by inverting the point-to-point map. We discuss that inverting the function or rewriting one description of motion to the other is not a pleasant experience. The chain rule is discussed which helps to relate Eulerian and Lagrangian descriptions of motion without inverting. For using the chain rule a new term — time derivative — is introduced and more specifically material derivative is discussed. The material time derivative is nothing but the time derivative with the reference position kept fixed. The concept of the material time derivative is applied to Eulerian acceleration and we see how it relates to Lagrangian acceleration. Also, while seeing the above relation explicit and implicit time dependence are seen. Finally, we introduce the chain rule, which relates these two descriptions of the motion.

In this video, the equation written for the relation between Eulerian and Lagrangian descriptions using the chain rule is discussed further and the equation is written in the form of gradients. The use of coordinate notations is shown for writing the same equation above and the equation is expressed as the sum of the partial time derivative and the convective time derivative. The equation is finally equal to the total or material time derivative. The application of the above equation is shown for the rigid body.