Derivation of Acceleration from Velocity with Partial derivatives

  • Thread starter fluidmech
  • Start date
  • #1
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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
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,260
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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).
 
  • #3
17
1
I see, I'm still a bit hazy on the mathematics of the partials, would you mind elaborating on that?
 
Last edited:
  • #5
17
1
That helped me tremendously. Now I understand it, thank you!
 

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