1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Problem about rate of change (multivariable calculus)

  1. Feb 23, 2013 #1
    1. The problem statement, all variables and given/known data
    Let T = f(x, y, z), where dx/dt = 4, dy/dt = 4 and dz/dt = -3
    Calculate dT/dt if dT/dx = 4, dT/dy = 7 and dT/dz = 9

    2. Relevant equations
    dT/dt = ∂T/∂x (dx/dt) + ∂T/∂y (dy/dt) + ∂T/∂z (dz/dt)

    3. The attempt at a solution
    I tried to get an explicit formula for T. I thought that, if dT/dx = 4, then T = 4x + c. Similarly, T = 7y + c and T = 9z + c. If i add the above expressions, I get:

    3T = 4x + 7y + 9z + C, so, T would be T = 4x/3 + 7y/3 + 3z + C, and I could use the chain rule and just substitute dx/dt and so on...

    Is my reasoning fine?
  2. jcsd
  3. Feb 24, 2013 #2
    Why do you need to do anything about T?

    You have the equation for dT/dt, and you know the values of everything on its right-hand side.
  4. Feb 24, 2013 #3


    User Avatar
    Science Advisor

    First, you can't get an explicit expression for T, there are infinitely many possibililties. But you don't need that, just the "chain rule"":
    [tex]\frac{dT}{dt}= \frac{\partial T}{\partial x}\frac{dx}{dt}+ \frac{\partial T}{\partial y}\frac{dy}{dt}+ \frac{\partial T}{\partial z}\frac{dz}{dt}[/tex]
  5. Feb 24, 2013 #4
    But I need ∂T/∂x and I have dT/dx = 4 (same for the others). Don't tell me ∂T/∂x = dT/dx, because I'm gonna cry
  6. Feb 24, 2013 #5
    Using d when you have a function with more than one variable is not correct notation. ∂T/∂x is the correct notation.
  7. Feb 24, 2013 #6
    Omg you are right. My teacher wrote it like dT/dx, and I never realized he wrote it wrong. Thank you for your help guys
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted