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Homework Help: Transport theorem, final integral

  1. Apr 12, 2012 #1


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    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


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    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)



    Nevermind, it is simply using the definition of the total derivative. That's all there is to it...
    Last edited: Apr 13, 2012
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