# Derivation of Acceleration from Velocity with Partial derivatives

## 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}##

- Matt

Dick
Homework Helper

## 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}##

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

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

Last edited:
Dick