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## 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.

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