Triple product rule

An informal derivation follows. Suppose that f(x, y, z) = 0. Write z as a function of x and y. Thus the total differential dz is

${\displaystyle dz=\left({\frac {\partial z}{\partial x}}\right)_{y}dx+\left({\frac {\partial z}{\partial y}}\right)_{x}dy}$

Suppose that we move along a curve with dz = 0, where the curve is parameterized by x. Thus y can be written in terms of x, so on this curve

${\displaystyle dy=\left({\frac {\partial y}{\partial x}}\right)_{z}dx}$

Therefore, the equation for dz = 0 becomes

${\displaystyle 0=\left({\frac {\partial z}{\partial x}}\right)_{y}\,dx+\left({\frac {\partial z}{\partial y}}\right)_{x}\left({\frac {\partial y}{\partial x}}\right)_{z}\,dx}$

Since this must be true for all dx, rearranging terms gives

${\displaystyle \left({\frac {\partial z}{\partial x}}\right)_{y}=-\left({\frac {\partial z}{\partial y}}\right)_{x}\left({\frac {\partial y}{\partial x}}\right)_{z}}$

Dividing by the derivatives on the right hand side gives the triple product rule

${\displaystyle \left({\frac {\partial x}{\partial y}}\right)_{z}\left({\frac {\partial y}{\partial z}}\right)_{x}\left({\frac {\partial z}{\partial x}}\right)_{y}=-1}$

Note that this proof makes many implicit assumptions regarding the existence of partial derivatives, the existence of the exact differential dz, the ability to construct a curve in some neighborhood with dz = 0, and the nonzero value of partial derivatives and their reciprocals. A formal proof based on mathematical analysis would eliminate these potential ambiguities.

The profile of a traveling wave at time t (solid line) and t+Δt (dashed line). In the time interval Δt, the point p₂ will rise up to the same height that p₁ had at time t.

A geometric realization of the triple product rule can be found in its close ties to the velocity of a traveling wave

${\displaystyle \phi (x,t)=A\cos(kx-\omega t)}$

shown on the right at time t (solid blue line) and at a short time later t+Δt (dashed). The wave maintains its shape as it propagates, so that a point at position x at time t will correspond to a point at position x+Δx at time t+Δt,

${\displaystyle A\cos(kx-\omega t)=A\cos(k(x+\Delta x)-\omega (t+\Delta t)).}$

This equation can only be satisfied for all x and t if kΔx-ωΔt=0, resulting in the formula for the phase velocity

${\displaystyle v={\frac {\Delta x}{\Delta t}}={\frac {\omega }{k}}.}$

To elucidate the connection with the triple product rule, consider the point p₁ at time t and its corresponding point (with the same height) p̄₁ at t+Δt. Define p₂ as the point at time t whose x-coordinate matches that of p̄₁, and define p̄₂ to be the corresponding point of p₂ as shown in the figure on the right. The distance Δx between p₁ and p̄₁ is the same as the distance between p₂ and p̄₂ (green lines), and dividing this distance by Δt yields the speed of the wave.

To compute Δx, consider the two partial derivatives computed at p₂,

${\displaystyle \left({\frac {\partial \phi }{\partial t}}\right)_{x}\Delta t={\text{rise from }}p_{2}{\text{ to }}{\bar {p}}_{1}{\text{ in time }}\Delta t{\text{ (gold line)}}}$

${\displaystyle \left({\frac {\partial \phi }{\partial x}}\right)_{t}={\text{slope of the wave (red line) at time }}t.}$

Dividing these two partial derivatives and using the definition of the slope (rise divided by run) gives us the desired formula for

${\displaystyle \Delta x=-{\frac {\left({\frac {\partial \phi }{\partial t}}\right)_{x}\Delta t}{\left({\frac {\partial \phi }{\partial x}}\right)_{t}}},}$

where the negative sign accounts for the fact that p₁ lies behind p₂ relative to the wave's motion. Thus, the wave's velocity is given by

${\displaystyle v={\frac {\Delta x}{\Delta t}}=-{\frac {\left({\frac {\partial \phi }{\partial t}}\right)_{x}}{\left({\frac {\partial \phi }{\partial x}}\right)_{t}}}.}$

For infinitesimal Δt, ${\displaystyle {\frac {\Delta x}{\Delta t}}=\left({\frac {\partial x}{\partial t}}\right)_{\phi }}$ and we recover the triple product rule

${\displaystyle v={\frac {\Delta x}{\Delta t}}=-{\frac {\left({\frac {\partial \phi }{\partial t}}\right)_{x}}{\left({\frac {\partial \phi }{\partial x}}\right)_{t}}}.}$

• Exact differential (has another derivation of the triple product rule)
• Total derivative
• Triple product for vectors and scalars.

• Elliott, J. R.; Lira, C. T. (1999). Introductory Chemical Engineering Thermodynamics (1st ed.). Prentice Hall. p. 184. ISBN 0-13-011386-7.
• Carter, Ashley H. (2001). Classical and Statistical Thermodynamics. Prentice Hall. p. 392. ISBN 0-13-779208-5.