Derivation of Acceleration from Velocity with Partial derivatives

[fluidmech]

Homework Statement

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.

Homework 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:

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

Thanks in advance!

- Matt

 

[Dick]

Homework Statement

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.

Homework 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:

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

Thanks in advance!

- Matt

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

 

[fluidmech]

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

 

[Dick]

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

Look up the chain rule for partial derivatives. E.g. http://mathworld.wolfram.com/ChainRule.html

 

[fluidmech]

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