(adsbygoogle = window.adsbygoogle || []).push({}); 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.)

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# Partial Derivative Proof (thermodynamics notation)

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