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Partial derivative

  1. Jul 7, 2012 #1

    soi

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    1. The problem statement, all variables and given/known data
    Hey, i ve got problem with a few partial derivative problems.

    1.I have a function T(x,t)
    Prove that dT/dt=∂T/∂t +∂T/∂x dx/dt

    2.Let u(x,y) and y(x,u) be continous, differentiable functions.
    Prove that
    ∂u/∂z=∂u/∂z ∂y/∂z

    3
    Let r(q1,q2,...qn) be a function of place depending on n coordinates.
    Show that ∂r/∂q=derivative of r/derivative of q
    ,
    2. Relevant equations



    3. The attempt at a solution
    Well, the first one i tried to prove chain rule for partial derivatives but i failed. I also cannot find one in the Web.

    Second comes from Leibniz notation for derivatives - but again I cant prove it myself or find a prove.

    Third- well i dont have a clue.
     
  2. jcsd
  3. Jul 7, 2012 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    In order that dT/dt exist you must be able to think of T as a function of t only. And since it is given as a function of both x and t, x itself must be a function of t. Now, one form of the chain rule for partial derivatives is
    [tex]\frac{df}{dt}= \frac{\partial f}{\partial x}\frac{dx}{dt}+ \frac{\partial f}{\partial y}\frac{dy}{dt}[/tex]
    where f is a function of x and y which are both functions of t. Replace y in that with t.

    Now, show us what you have tried on the others.
     
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