# Prove that $\frac{\partial x}{\partial y} \frac{\partial y}{\partial z} \frac{\partial z}{\partial x} = -1$ and verify ideal gas law

Ok guys, continuing my passage through edwards... here is the question... thanks for hints/solutions in advance:

Suppose $f(x,y,z)=0$ can be solved for each of the three variables $x,y,z$ as a differentiable function of the other two. Then prove that

$\displaystyle \frac{\partial x}{\partial y} \frac{\partial y}{\partial z} \frac{\partial z}{\partial x} = -1$

Verify this is the case for the ideal gas equation $pv =RT$ where (where $p,v,T$ are the three variables and $R$ is the constant).

• Use the implicit function theorem. Mar 29, 2013 at 3:55
• math.stackexchange.com/questions/68039/… Mar 29, 2013 at 4:19
• Triple product rule en.wikipedia.org/wiki/Triple_product_rule Mar 29, 2013 at 4:23
• thanks guys... that helps a lot.
– Neo
Mar 29, 2013 at 4:27
• You could also look up the Upstairs-Downstairs, Inside-Out Formula, Mar 29, 2013 at 4:35

Define $f(x(y,z),y,z)=f(x,y(x,z),z)=f(x,y,z(x,y))=0$. Then by the chain rule, \begin{eqnarray*} \frac{\partial f}{\partial y}+\frac{\partial f}{\partial x}\frac{\partial x}{\partial y} & = & 0\\ \frac{\partial f}{\partial z}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial z} & = & 0\\ \frac{\partial f}{\partial x}+\frac{\partial f}{\partial z}\frac{\partial z}{\partial x} & = & 0 \end{eqnarray*} So that \begin{eqnarray*} \frac{\partial x}{\partial y} & = & -\frac{\frac{\partial f}{\partial y}}{\frac{\partial f}{\partial x}}\\ \frac{\partial y}{\partial z} & = & -\frac{\frac{\partial f}{\partial z}}{\frac{\partial f}{\partial y}}\\ \frac{\partial z}{\partial x} & = & -\frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial z}} \end{eqnarray*} Hence, $$\frac{\partial x}{\partial y}\frac{\partial y}{\partial z}\frac{\partial z}{\partial x}=-\frac{\frac{\partial f}{\partial y}}{\frac{\partial f}{\partial x}}\frac{\frac{\partial f}{\partial z}}{\frac{\partial f}{\partial y}}\frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial z}}=-1.$$