# Partial Derivative Proof (thermodynamics notation)

• Sep 2nd 2008, 09:15 AM
Jacobpm64
Partial Derivative Proof (thermodynamics notation)
Show that: $\left(\frac{\partial z}{\partial y}\right)_{u} = \left(\frac{\partial z}{\partial x}\right)_{y} \left[ \left(\frac{\partial x}{\partial y}\right)_{u} - \left(\frac{\partial x}{\partial y}\right)_{z} \right]$

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

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:
$\left(\frac{\partial x}{\partial y}\right)_{z} \left(\frac{\partial y}{\partial z}\right)_{x} \left(\frac{\partial z}{\partial x}\right)_{y} = -1$

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:
$\left(\frac{\partial z}{\partial y} \right)_{u} = \left(\frac{\partial z}{\partial x} \right)_{u} \left(\frac{\partial x}{\partial y}\right)_{u}$

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

Any help would be appreciated.

(This is for a thermodynamics course, but we are still in the mathematics introduction.)
• Sep 2nd 2008, 04:30 PM
i too am an engineer but i only took it to the undergrad lvl and am not familiar with this notation... can you clarify?
• Sep 2nd 2008, 04:33 PM
Jacobpm64
Sure thing.

If I say something like $\left(\frac{\partial z}{\partial x}\right)_{y}$.

This means... The partial derivative of z with respect to x, holding y constant.
• Sep 2nd 2008, 04:35 PM
ThePerfectHacker
Quote:

Originally Posted by Jacobpm64
Sure thing.

If I say something like $\left(\frac{\partial z}{\partial x}\right)_{y}$.

This means... The partial derivative of z with respect to x, holding y constant.

There is no need to write,
$\left(\frac{\partial z}{\partial x}\right)_{y}$

Because the notation,
$\frac{\partial z}{\partial x}$
Means exactly that i.e. "you hold all variables constant except x".
• Sep 2nd 2008, 04:38 PM
Jacobpm64
Oh, i thought it was different in thermodynamics, Please excuse me.
• Sep 2nd 2008, 04:41 PM