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When do total differentials cancel with partial derivatives

  1. Dec 25, 2015 #1
    I've just done a derivation and had to use the following

    [itex]u_{b}u^{c}\partial_{c}\rho = u_{b}\frac{dx^{c}}{d\tau}\frac{\partial\rho}{\partial x^{c}} = u_{b}\frac{d\rho}{d\tau}[/itex]

    We've done this cancellation a lot during my GR course, but I'm not clear exactly when/why this is possible.

    EDIT: is this only true in inertial coordinates?
     
  2. jcsd
  3. Dec 25, 2015 #2
    Are you familiar with the multivariable chain rule
    [tex]\frac{d f (x,y)}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}?[/tex]
    The 'cancellation' you performed there is simply a simplification using the chain rule (remember that you are using the Einstein summation convention).
     
  4. Dec 25, 2015 #3
    Thanks, i did notice that of course after posting o:)
     
  5. Dec 27, 2015 #4

    HallsofIvy

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    And, while it may be a useful "mnemonic", the derivative, ordinary or partial, is NOT a fraction and the "chain rule" does NOT involve "cancelling".
     
  6. Dec 27, 2015 #5

    bcrowell

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