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 = 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 Here the subscripts indicate which variables are held constant when the partial derivative is taken. That is, to explicitly compute the partial derivative of xwith respect toy 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.
Derivation
An informal derivation follows. Suppose that f = 0. Write z as a function of x and y. Thus the total differentialdz 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 differentialdz, 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 f=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
Applications
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 and at a short time later t+Δt. 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 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, 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 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
