Register to reply

Partial Derivative Proof (thermodynamics notation)

Share this thread:
Jacobpm64
#1
Sep2-08, 01:23 AM
P: 239
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.)
Phys.Org News Partner Science news on Phys.org
Flapping baby birds give clues to origin of flight
Prions can trigger 'stuck' wine fermentations, researchers find
Socially-assistive robots help kids with autism learn by providing personalized prompts

Register to reply

Related Discussions
Converting partial derivative w.r.t. T to partial derivative w.r.t. 1/T Calculus & Beyond Homework 2
Replacing total derivative with partial derivative in Griffiths' book Advanced Physics Homework 3
Partial derivative notation Calculus & Beyond Homework 6
Confused on this notation! partial derivatives! Calculus & Beyond Homework 2
Total derivative -> partial derivative Differential Equations 6