Partial Derivative Proof (thermodynamics notation)

  1. 1. The problem statement, all variables and given/known data
    Show that: [tex] \left(\frac{\partial z}{\partial y}\right)_{u} = \left(\frac{\partial z}{\partial x}\right)_{y} \left[ \left(\frac{\partial x}{\partial y}\right)_{u} - \left(\frac{\partial x}{\partial y}\right)_{z} \right] [/tex]

    2. Relevant equations
    I have Euler's chain rule and "the splitter." Also the property, called the "inverter" where you can reciprocate a partial derivative.

    3. The attempt at a solution
    If I write Euler's chain rule, I only know how to write it when there are 3 variables, I usually write it in the form:
    [tex]\left(\frac{\partial x}{\partial y}\right)_{z} \left(\frac{\partial y}{\partial z}\right)_{x} \left(\frac{\partial z}{\partial x}\right)_{y} = -1 [/tex]

    Where I can write x,y,z in any order as long as each variable is used in every spot. However, I do not know how to work this chain rule if I have an extra variable (u in this case).

    I also tried using the "splitter" to do something like writing:
    [tex] \left(\frac{\partial z}{\partial y} \right)_{u} = \left(\frac{\partial z}{\partial x} \right)_{u} \left(\frac{\partial x}{\partial y}\right)_{u} [/tex]

    However, I do not know what to do with this because I have the term
    [tex] \left(\frac{\partial z}{\partial x} \right)_{u} [/tex] , which doesn't appear in the original problem.

    Any help would be appreciated.

    Thanks in advance.

    (This is for a thermodynamics course, but we are still in the mathematics introduction.)
  2. jcsd
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