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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

Note: In each factor the variable in the numerator is considered to be an implicit function of the other two. In each factor the subscripted variable is being held constant.

Here the subscripts indicate which variables are held constant when the partial derivative is taken. That is, to explicitly compute the partial derivative of x with respect to y with z held constant, one would write x as a function of y and z and take the partial derivative of this function with respect to y only.

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,

Various other forms of the rule are present in the literature; these can be derived by permuting the variables {x, y, z}.

Contents

DerivationEdit

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

 

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

 

Therefore, the equation for dz = 0 becomes

 

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

 

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

 

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 derivationEdit

Suppose a function f(x,y,z)=0, where x,y and z are functions of each other. Write the total differentials of the variables

 
 

Substitute dy into dx

 

By using the chain rule one can show the coefficient of dx is equal to one, thus the coefficient of dz must be zero

 

Subtracting the second term and multiplying by its inverse gives the triple product rule

 

ApplicationsEdit

 
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

 

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,

 

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

 

To elucidate the connection with the triple product rule, consider the point p1 at time t and its corresponding point (with the same height) 1 at t+Δt. Define p2 as the point at time t whose x-coordinate matches that of 1, and define 2 to be the corresponding point of p2 as shown in the figure on the right. The distance Δx between p1 and 1 is the same as the distance between p2 and 2 (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,

 
 

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

 

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

 

For infinitesimal Δt,   and we recover the triple product rule

 

See alsoEdit

ReferencesEdit

  • Elliott, JR, and Lira, CT. Introductory Chemical Engineering Thermodynamics, 1st Ed., Prentice Hall PTR, 1999. p. 184.
  • Carter, Ashley H. Classical and Statistical Thermodynamics, Prentice Hall, 2001, p. 392.