1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Transport theorem, final integral

  1. Apr 12, 2012 #1

    Leb

    User Avatar

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

    Show that

    [itex]\frac{d}{dt}\int \rho r^{2}\phi dr = \int \rho r^{2}\frac{d\phi}{dr} dr [/itex]

    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
    [itex]\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 [/itex]

    Note: Integrals are evaluated from a to b, and v(x,t) = dx/dt (e.g. da/dt = v(a,t))
     
  2. jcsd
  3. Apr 12, 2012 #2
    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?
     
  4. Apr 13, 2012 #3

    Leb

    User Avatar

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

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

    25qrxp4.jpg

    EDIT:

    Nevermind, it is simply using the definition of the total derivative. That's all there is to it...
     
    Last edited: Apr 13, 2012
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Transport theorem, final integral
  1. Parallel transport (Replies: 0)

Loading...