Understanding why cyclic rule works

[gloryofgreece]

I don't quite understand why cyclic rule works (from Pchem)

(del x/ del y)_z = part of x with respect to y, hold z constant

I don't know why is it negative 1?

del x/ del y)_z * del y/ del z)_x * del z/ del x)_y = -1

 

[Gerenuk]

You could derive it. Not sure if there is any other way to imagine it
$$\mathrm{d}z=\left.\frac{\partial z}{\partial x}\right)_y\mathrm{d}x+\left.\frac{\partial z}{\partial y}\right)_x\mathrm{d}y$$
Now to get $\left.\frac{\partial x}{\partial y}\right)_z$ from the above equation just find the fraction $\frac{\mathrm{d}x}{\mathrm{d}y}$ under the condition that $\mathrm{d}z=0$, i.e.
$$\left.\frac{\partial x}{\partial y}\right)_z=\left.\frac{\mathrm{d} x}{\mathrm{d} y}\right)_{\mathrm{d}z=0}=-\frac{\left.\partial z/\partial y\right)_x}{\left.\partial z/\partial x\right)_y}$$
The minus sign comes from rearranging the first equation.

Also
$$\left.\frac{\partial z}{\partial y}\right)_x=\frac{1}{\left.\frac{\partial y}{\partial z}\right)_x}$$
or
$$\left.\frac{\partial z}{\partial y}\right)_x\left.\frac{\partial y}{\partial z}\right)_x=1$$
The apparent cancellation is only possible since both derivatives have the same variables kept constant!