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I Evaluate using Leibniz rule and/or chain rule

  1. Dec 6, 2016 #1
    I want to evaluate
    $$ \frac{d}{dt}\int_{0}^{^{\eta(t)}}\rho(p,t)dz $$

    where p itself is $$ p=p(z,t) $$

    I have the feeling I have to use Leibniz rule and/or chain rule, but I'm not sure how...

    Thanks.
     
  2. jcsd
  3. Dec 6, 2016 #2

    fresh_42

    Staff: Mentor

    So ##\rho(p,t)=\rho(p(z,t),t)=\varphi(z,t)##, and if ##\phi_z(z,t)## is the anti-derivative of ##\varphi(z,t)## with respect to the first coordinate, your are actually looking for ##\frac{d}{dt} \left( \phi_z(\eta(t),t) - \phi_z(0,t) \right)##.
    Don't know whether this helps.
     
  4. Dec 6, 2016 #3

    lurflurf

    User Avatar
    Homework Helper

    The chain rule lets us differentiate each instance of the variable separately.
    $$\dfrac{d}{dt}\int_{0}^{^{\eta(t)}}\rho(p(z,t),t)dz=\left.\left(\dfrac{\partial}{\partial u}+\dfrac{\partial}{\partial v}+\dfrac{\partial}{\partial w}\right)\int_{0}^{^{\eta(u)}}\rho(p(z,v),w)dz\right|_{u,v,w=t}$$
    So replace each t by a dummy variable differentiate with respect to each and add them up, then replace each with t again.
     
    Last edited: Dec 6, 2016
  5. Dec 8, 2016 #4
    Thanks fresh_24 and lurflurf for your quick and helfpul answers! Have a nice day.
     
  6. Dec 10, 2016 #5

    lavinia

    User Avatar
    Science Advisor

    $$ F(t) = \int_{0}^{^{t}}\rho(p,t)dz $$ is a function of ##t##.

    Your function is ##F(η(t))##.
     
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