Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Where did this equation come from?

  1. Mar 16, 2007 #1
    1. The problem statement, all variables and given/known data
    This equation in the attached document appeared in thermal physics as a mathematical identity. I like to know mathematically how it is derived.

    3. The attempt at a solution
    I don't know where to start

    Attached Files:

  2. jcsd
  3. Mar 16, 2007 #2
    I can't open the gif file. Maybe others can.
  4. Mar 16, 2007 #3
    A Mentor needs to approve it. They're probably catching up on their Z's, now. :biggrin:
  5. Mar 17, 2007 #4


    User Avatar
    Science Advisor

    That's just the chain rule for partial derivatives- the notation, being physics rather than mathematics is a little peculiar. The subscripts mean "this variable being treated as a constant.
  6. Mar 17, 2007 #5
    You are right in that the notations are not clear.
    I assume that the 4 variables x,y,z,w are dependent variables? But what are the independent variables? Do I need to use the Jacobian?
  7. Mar 17, 2007 #6


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    On the left side, x is a function of y and z, x(y,z).

    Then imagine rewriting this as a function of y and w instead, where w is some function of *both* y and z. The only condition is that the resulting function x(y,w) does not depend on z explicitly (but it does implicitly through the dependence of w on z).

    In other words, one goes from x(y,z) to x(y,w(y,z)).

    Edit: when I say that w is a function of both y and z, I mean that it *may* be a function of both y and z. Of course, a special and trivial case is w=z. A slightly more general case is w is some function of z. In both cases, obviously the partial derivative of x with respect to y is the same no matter if x is expressed in terms of y,z or in terms of y,w. But if w is a function of both z and y, the formula needed is the one you quoted.

    Last edited: Mar 17, 2007
  8. Mar 18, 2007 #7
    I see, how did you work it out?

    They should state
    if x(y,z)=x(y,w(y,z)) then the equation I showed.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook