Partial Derivative Proof (thermodynamics notation)


by Jacobpm64
Tags: derivative, notation, partial, proof, thermodynamics
Jacobpm64
Jacobpm64 is offline
#1
Sep2-08, 01:23 AM
P: 240
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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