# Geometrical interpretation of the Triple Product Rule

Most of us who have done multivariable calculus are familiar with the rule $$\left( \frac{\partial x}{\partial y} \right)_{z} \left( \frac{\partial y}{\partial z} \right)_{x} \left( \frac{\partial z}{\partial x} \right)_{y} = -1,$$ particularly useful in thermodynamics. The normal way to prove this (as shown in either of these Wikipedia articles is by mucking about with the algebra of differentials and substituting, and there's nothing as geometrically unintuitive as pushing algebraic symbols around. The $-1$ is also famously counterintuitive to people seeing it for the first time.

Therefore, is there a nice intuitive geometrical derivation or interpretation of this result? (And, indeed, its more-variabled generalisations?)

• This is a great geometric explanation. Indeed, for any plane $ax+by+cz+d=0$ with $a$, $b$ and $c$ all nonzero, a direct computation shows that the product of the slopes of its intersections with the coordinate planes, suitably oriented, is $-1$. – amd Apr 7 '18 at 1:41