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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


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    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


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    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.
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