1. ## Material derivative

How did this guy: https://www.youtube.com/watch?v=BLVY69SYVBQ at 6:27 get the right hand side expression?

He said: When we apply chain-rule, but I still don't get.
Thank you.

2. ## Re: Material derivative

So no one understands material derivative? Maybe I should ask on physics forum, because this is a physics topic.

3. ## Re: Material derivative

It's the chain rule for multiple variables - if f is a function of two variables x and y, each of which is a function of t, then:

$\frac{df}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}$

Changes in t cause changes in both x and y, which each cause changes in f, and the effects are additive.

To understand what follows later, it might help to break r into x, y, and z. Then you'd have 4 terms, but the x, y, and z terms should combine to that $v \cdot \nabla C$.

- Hollywood

4. ## Re: Material derivative

Originally Posted by Nforce
So no one understands material derivative? Maybe I should ask on physics forum, because this is a physics topic.
A "material" derivative can have several meanings in Physics but it is still the same as a derivative in Mathematics.

-Dan

5. ## Re: Material derivative

Originally Posted by hollywood
It's the chain rule for multiple variables - if f is a function of two variables x and y, each of which is a function of t, then:

$\frac{df}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}$

Changes in t cause changes in both x and y, which each cause changes in f, and the effects are additive.

To understand what follows later, it might help to break r into x, y, and z. Then you'd have 4 terms, but the x, y, and z terms should combine to that $v \cdot \nabla C$.

- Hollywood
Yes but by definition we get also ordinary derivative and he got 3 partial derivatives and no ordinary one. Can someone write down step by step, how do we get then the expression?

6. ## Re: Material derivative

Sorry, but is there no one who has time to write down step by step derivation?

7. ## Re: Material derivative

Can anyone please derive it? I don't know how to, this really isn't a homework problem, I want to see and understand.

8. ## Re: Material derivative

Originally Posted by Nforce
Can anyone please derive it? I don't know how to, this really isn't a homework problem, I want to see and understand.
See here. Especially the header "Total Derivative as a Differential Form."

-Dan