How do I deduce some basic thermodynamic identities using multivariate calculus?

[CharlieCW]

Homework Statement

Let x, y and z satisfy the state function $f(x, y, z) = 0$ and let $w$ be a function of only two of these variables. Show the following identities:

$$\left(\frac{\partial x}{\partial y}\right )_w \left(\frac{\partial y}{\partial z}\right )_w =\left(\frac{\partial x}{\partial z}\right )_w$$

$$\left(\frac{\partial x}{\partial y}\right )_w=\frac{1}{\left(\frac{\partial x}{\partial y}\right )_w}$$

$$\left(\frac{\partial x}{\partial y}\right )_z \left(\frac{\partial y}{\partial z}\right )_y \left(\frac{\partial z}{\partial x}\right )_x =-1$$

$$\left(\frac{\partial x}{\partial y}\right )_z=\left(\frac{\partial x}{\partial y}\right )_w + \left(\frac{\partial x}{\partial w}\right )_y \left(\frac{\partial w}{\partial}\right )_z$$

Homework Equations

All relationships must be deduced only by assuming $f(x,y,z)=0$ and $w$ function only of two of these variables. Cannot introduce topology or complex analysis theorems (multivariate calculus basics (i.e. chain rule, exact differentials) are allowed).

The Attempt at a Solution

These are some thermodynamic identities that I've been using for a while, and while I have seen the proofs of some of this in GR and QFT (for example, the inverse function theorem), I'm struggling a bit on how deduce them from more basic calculus theorems used in statistical physics.

To begin with, I noted that since I have $f(x,y,z)=0$ then I can write, for example, $x=x(y,z)$. Moreover, assuming $w=w(x,y)$, I can also write $w=w(x(y,z),z)=w=(y,z)$, and so on.

For part (a), I used the chain rule to show straightforward that:

$$\left(\frac{\partial x}{\partial z}\right )_w=\left(\frac{\partial x}{\partial y}\right )_w \left(\frac{\partial y}{\partial z}\right )_w$$

However, I'm not entirely sure if this is an acceptable proof or if there's a way to develop it a bit more.

For part (b), I tried, for example:

$$\left ( \frac{\partial x}{\partial y} \right )_w=\left ( \frac{\partial x}{\partial z} \right )_w \left ( \frac{\partial z}{\partial y} \right )_w=\left ( \frac{\partial x}{\partial y} \right )_w \left ( \frac{\partial y}{\partial z} \right )_w \left ( \frac{\partial z}{\partial y} \right )_w$$

Were I applied the chain rule again to the left term. The above expression reduces to:

$$1=\left ( \frac{\partial y}{\partial z} \right )_w \left ( \frac{\partial z}{\partial y} \right )_w \ \ \ \longrightarrow \ \ \ \left ( \frac{\partial y}{\partial z} \right )_w=\frac{1}{\left ( \frac{\partial z}{\partial y} \right )_w}$$

I have seen several proofs online using complex analysis or topology, so I'm not sure if this derivation is also valid or I just forced it.

For part (c), the proof is pretty straightforward using differentials and the inverse function theorem, so I'll skip it (a simple proof can be found here: https://en.wikipedia.org/wiki/Triple_product_rule)

For part (d), I'm having the most trouble. I tried using differentials and the chain rule:

$$\left ( \frac{\partial x}{\partial y} \right )_z=\left ( \frac{\partial x}{\partial w} \right )_z \left ( \frac{\partial w}{\partial y} \right )_z$$

$$dy=\left ( \frac{\partial y}{\partial x} \right )_z dx + \left ( \frac{\partial y}{\partial z} \right )_x dz$$

$$dx=\left ( \frac{\partial x}{\partial y} \right )_z dy + \left ( \frac{\partial x}{\partial z} \right )_y dz$$

Substituting the first and second expressions into the third one:

$$dx=\left ( \frac{\partial x}{\partial w} \right )_z \left ( \frac{\partial w}{\partial y} \right )_z \left ( \left ( \frac{\partial y}{\partial x} \right )_z dx + \left ( \frac{\partial y}{\partial z} \right )_x dz \right ) + \left ( \frac{\partial x}{\partial z} \right )_y dz$$

The coefficient for $dx$ should be equal to $1$, while the coefficient for $dz$ should be zero. Therefore, we get the following equalities:

$$\left ( \frac{\partial x}{\partial w} \right )_z \left ( \frac{\partial w}{\partial y} \right )_z \left ( \frac{\partial y}{\partial x} \right )_z=1$$

$$\left ( \frac{\partial x}{\partial z} \right )_y=-\left ( \frac{\partial x}{\partial w} \right )_z \left ( \frac{\partial w}{\partial y} \right )_z \left ( \frac{\partial y}{\partial z} \right )_x$$

However, this doesn't seem to lead anywhere, so this is probably the one I have the least idea how to proceed.

 

[CptXray]

I think I've seen these in Callen's "Thermodynamics and Introduction to Thermostatics" in the appendix. Although, I'm not sure if (d) can be found there.

 

[CharlieCW]

I think I've seen these in Callen's "Thermodynamics and Introduction to Thermostatics" in the appendix. Although, I'm not sure if (d) can be found there.

Thanks, I'll check if we got it in the library after the morning lectures and I'll update you if I find the solution.

Edit: I checked the book, they have the proofs I needed. As for the last one, I found it here in page 4: http://mutuslab.cs.uwindsor.ca/schurko/introphyschem/handouts/mathsht.pdf