Partial Derivative Proof (thermodynamics notation)

[Jacobpm64]

Homework Statement

Show that: $$\left(\frac{\partial z}{\partial y}\right)_{u} = \left(\frac{\partial z}{\partial x}\right)_{y} \left[ \left(\frac{\partial x}{\partial y}\right)_{u} - \left(\frac{\partial x}{\partial y}\right)_{z} \right]$$

Homework Equations

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

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

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.

Thanks in advance.

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

 

[mighty2000]

Thank you for your post. I am happy to help you with this problem.

First, let's define some notation to make things clearer. Let z(x, y, u) be a function of three variables x, y, and u. We can rewrite the given equation as:

\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]

As you mentioned, Euler's chain rule allows us to write:

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

But this is not the same as the given equation. In order to use the chain rule, we need to find a way to incorporate the extra variable u into the equation. This is where the "splitter" comes in.

We can use the "splitter" to write:

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

Substituting this into our original equation, we get:

\left(\frac{\partial z}{\partial y}\right)_{u} = \left(\frac{\partial z}{\partial x}\right)_{u} \left(\frac{\partial x}{\partial y}\right)_{u} \left[ \left(\frac{\partial x}{\partial y}\right)_{u} - \left(\frac{\partial x}{\partial y}\right)_{z} \right]

Now, we can use the "inverter" property to write:

\left(\frac{\partial x}{\partial y}\right)_{u} = \frac{1}{\left(\frac{\partial y}{\partial x}\right)_{u}}

Substituting this into our equation, we get:

\left(\frac{\partial z