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Derivation of Acceleration from Velocity with Partial derivatives

  1. Nov 7, 2012 #1
    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}##

    Thanks in advance!

    - Matt
     
  2. jcsd
  3. Nov 7, 2012 #2

    Dick

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    You want to think of V as a function of four variables V(t,x,y,z).
     
  4. Nov 7, 2012 #3
    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
  5. Nov 7, 2012 #4

    Dick

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  6. Nov 7, 2012 #5
    That helped me tremendously. Now I understand it, thank you!
     
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