# Transport theorem, final integral

1. Apr 12, 2012

### Leb

1. The problem statement, all variables and given/known data

Show that

$\frac{d}{dt}\int \rho r^{2}\phi dr = \int \rho r^{2}\frac{d\phi}{dr} dr$

2. Relevant equations
Fundamental theorem of calculus

3. The attempt at a solution

So I follow the derivation from the textbook and I think I get the rather sneaky rearrangement of the derivatives, but I do not see how
$\int \rho r^{2}\frac{d\phi}{dt} dr = \int \rho r^{2}\left(\frac{\partial \phi}{\partial t}+v\frac{\partial \phi}{\partial r}\right)dr$

Note: Integrals are evaluated from a to b, and v(x,t) = dx/dt (e.g. da/dt = v(a,t))

2. Apr 12, 2012

### Mindscrape

What do the variables depend on? I don't see any problem with the line given a t and r dependence in phi. What subject is this in? Care to share some more problem details?

3. Apr 13, 2012

### Leb

Sorry, I thought it was something standard. It is a mechanics course.

t-time, r - radius, $\rho (r,t)$ is density, $\phi (r,t)$ is an arbitrary differentiable function, a=a(t), b=b(t)

EDIT:

Nevermind, it is simply using the definition of the total derivative. That's all there is to it...

Last edited: Apr 13, 2012