Triple product rule

The triple product rule, known variously as the cyclic chain rule, cyclic relation, cyclical rule or Euler's chain rule, is a formula which relates partial derivatives of three interdependent variables. The rule finds application in thermodynamics, where frequently three variables can be related by a function of the form ''f''(''x'', ''y'', ''z'') = 0, so each variable is given as an implicit function of the other two variables. For example, an equation of state for a fluid relates temperature, pressure, and volume in this manner. The triple product rule for such interrelated variables ''x'', ''y'', and ''z'' comes from using a reciprocity relation on the result of the implicit function theorem, and is given by
:$\left(\frac\right)\left(\frac\right)\left(\frac\right) = -1,$
where each factor is a partial derivative of the variable in the numerator, considered to be a function of the other two.

The advantage of the triple product rule is that by rearranging terms, one can derive a number of substitution identities which allow one to replace partial derivatives which are difficult to analytically evaluate, experimentally measure, or integrate with quotients of partial derivatives which are easier to work with. For example,

:$\left(\frac\right) = - \frac$

Various other forms of the rule are present in the literature; these can be derived by permuting the variables .

Derivation

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

:$dz = \left(\frac\right)dx + \left(\frac\right) dy$

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

:$dy = \left(\frac\right) dx$

Therefore, the equation for ''dz'' = 0 becomes

:$0 = \left(\frac\right) \, dx + \left(\frac\right) \left(\frac\right) \, dx$

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

:$\left(\frac\right) = -\left(\frac\right) \left(\frac\right)$

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

:$\left(\frac\right)\left(\frac\right) \left(\frac\right) = -1$

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.

Alternative derivation

Suppose a function , where , , and are functions of each other. Write the total differentials of the variables
$dx = \left(\frac\right) dy + \left(\frac\right) dz$
$dy = \left(\frac\right) dx + \left(\frac\right) dz$
Substitute into
dx = \left(\frac\right) \left \left(\frac\right) dx + \left(\frac\right) dz\right+ \left(\frac\right) dz
By using the chain rule one can show the coefficient of on the right hand side is equal to one, thus the coefficient of must be zero
$\left(\frac\right) \left(\frac\right) + \left(\frac\right) = 0$
Subtracting the second term and multiplying by its inverse gives the triple product rule
$\left(\frac\right) \left(\frac\right) \left(\frac\right) = -1.$

Applications

Example: Ideal Gas Law

The ideal gas law relates the state variables of pressure (P), volume (V), and temperature (T) via
:$PV=nRT$

which can be written as
:$f(P,V,T) = PV-nRT = 0$

so each state variable can be written as an implicit function of the other state variables:
:
\begin
P &= P(V,T) = \frac \\ emV &= V(P,T) = \frac \\ emT &= T(P,V) = \frac
\end

From the above expressions, we have
:
\begin
-1 &= \left( \frac \right) \left( \frac \right) \left( \frac \right) \\ em&= \left( -\frac \right) \left( \frac \right) \left( \frac \right) \\ em&= \left( -\frac \right) \\ em& = -\frac = -1
\end

Geometric Realization

A geometric realization of the triple product rule can be found in its close ties to the velocity of a traveling wave
:$\phi(x,t) = A \cos (kx - \omega t)$
shown on the right at time ''t'' (solid blue line) and at a short time later ''t''+Δ''t'' (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''+Δ''x'' at time ''t''+Δ''t'',
:$A \cos (kx - \omega t) = A \cos (k (x + \Delta x) - \omega (t + \Delta t)).$
This equation can only be satisfied for all ''x'' and ''t'' if , resulting in the formula for the phase velocity
:$v = \frac = \frac.$

To elucidate the connection with the triple product rule, consider the point ''p''₁ at time ''t'' and its corresponding point (with the same height) ''p̄''₁ at ''t''+Δ''t''. Define ''p''₂ as the point at time ''t'' whose x-coordinate matches that of ''p̄''₁, and define ''p̄''₂ to be the corresponding point of ''p''₂ as shown in the figure on the right. The distance Δ''x'' between ''p''₁ and ''p̄''₁ is the same as the distance between ''p''₂ and ''p̄''₂ (green lines), and dividing this distance by Δ''t'' yields the speed of the wave.

To compute Δ''x'', consider the two partial derivatives computed at ''p''₂,
:$\left( \frac \right) \Delta t = \textp_2\text\bar_1\text\Delta t\text$
:$\left( \frac \right) = \textt.$
Dividing these two partial derivatives and using the definition of the slope (rise divided by run) gives us the desired formula for
:$\Delta x = - \frac,$
where the negative sign accounts for the fact that ''p''₁ lies behind ''p''₂ relative to the wave's motion. Thus, the wave's velocity is given by
:$v = \frac = - \frac.$
For infinitesimal Δ''t'', $\frac = \left( \frac \right)$ and we recover the triple product rule
:$v = \frac = - \frac.$

See also

*
* (has another derivation of the triple product rule)
*
*
* and scalars.

References

*
* {{cite book , last=Carter , first=Ashley H. , title=Classical and Statistical Thermodynamics , publisher=Prentice Hall , year=2001 , page=392 , isbn=0-13-779208-5

Articles containing proofs
Laws of thermodynamics
Multivariable calculus
Theorems in analysis
Theorems in calculus