First This is what I have saw in a thermodynamics textbook:
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"There are three homogeneous system,and their parameters of the equation of state:
$O=f(x_1,\ldots,x_n)$
$P=g(y_1,\ldots,y_m)$
$Q=h(z_1,\ldots,z_k)$
If $O=P$, there must be a function $F$,$F(x_1,\ldots,x_n;y_1,\ldots,y_m)=0$ So that,if $P=Q$ and $O=Q$,then:
$G(x_1,\ldots,x_n;z_1,\ldots,z_k)=0$
$H(y_1,\ldots,y_n;z_1,\ldots,z_k)=0$
Let $\frac{∂F}{x_1}≠0$ and $\frac{∂G}{x_1}≠0$,then From $F(x_1,\ldots,x_n;y_1,\ldots,y_m)=0$ and $G(y_1,\ldots,y_n;z_1,\ldots,z_k)=0$ we can find $x_1$:
$x_1=F_1(y_1,\ldots,y_m;x_2,\ldots,x_n)=0$
$x_1=G_1(z_1,\ldots,z_k;x_2,\ldots,x_n)=0$
$F_1=G_1$
In this case,$H(y_1,\ldots,y_m;z_1,\ldots,z_k)=0$ if and only if $F_1$ and $G_1$ is this form:
$F_1=a(x_2,\ldots,x_n)s(y_1,\ldots,y_m)+b(x_2,\ldots,x_n)$
$G_1=a(x_2,\ldots,x_n)r(z_1,\ldots,z_k)+b(x_2,\ldots,x_n)$
Let $f(x_1,\ldots,x_n)=\frac{x_1-b(x_2,\ldots,x_n)}{a(x_2,\ldots,x_n)}$,then:
$f(x_1,\ldots,x_n)=s(y_1,\ldots,y_m)=r(z_1,\ldots,z_k)$
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So I have some questions about this discussion:
① why $x_1=F_1(y_1,\ldots,y_m;x_2,\ldots,x_n)=G_1(z_1,\ldots,z_k;x_2,\ldots,x_n)=0$,when $\frac{∂F}{x_1}≠0$ and $\frac{∂G}{x_1}≠0$?
② why $F_1$ and $G_1$ have to be that form in the discussion?
③why we can let $f(x_1,\ldots,x_n)=\frac{x_1-b(x_2,\ldots,x_n)}{a(x_2,\ldots,x_n)}$,why we have to do this?
④Is $g(y_1,\ldots,y_m)=s(y_1,\ldots,y_m)$ and $h(z_1,\ldots,z_k)=r(z_1,\ldots,z_k)$,in other words,is $s(y_1,\ldots,y_m)$ and $r(z_1,\ldots,z_k)$ are the equation of state of $P$ and $Q$?
I have post this question on mathoverflow section now (https://mathoverflow.net/questions/327369/questions-about-using-mathematical-methods-to-prove-the-caratheodorys-concept-o)