# Derivation of Acceleration from Velocity with Partial derivatives

1. Nov 7, 2012

### fluidmech

1. The problem statement, all variables and given/known data
I'm taking a fluid mechanics class and I'm having an issue with acceleration and background knowledge. I know this is ridiculous, but I was hoping someone might be able to explain it for me.

2. Relevant equations
I definitely understand:
$a=\frac{d\vec{V}}{dt}$

And I know that u, v, and w are components of the velocity, $\vec{V}=<u,v,w>$

But how do I use the chain rule of differentiation to get to:

$\vec{a}=\frac{d\vec{V}}{dt}=\frac{\partial \vec{V}}{\partial t} +\frac{\partial \vec{V}}{\partial x}\frac{dx}{dt} +\frac{\partial \vec{V}}{\partial y}\frac{dy}{dt} +\frac{\partial \vec{V}}{\partial z}\frac{dz}{dt}$

- Matt

2. Nov 7, 2012

### Dick

You want to think of V as a function of four variables V(t,x,y,z).

3. Nov 7, 2012

### fluidmech

I see, I'm still a bit hazy on the mathematics of the partials, would you mind elaborating on that?

Last edited: Nov 7, 2012
4. Nov 7, 2012

5. Nov 7, 2012

### fluidmech

That helped me tremendously. Now I understand it, thank you!