page_title: Triple product rule
section_title: Applications
hierarchical_section_title: Triple product rule / Applications
caption_attribution_description: English: 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.
context_page_description: 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
  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.
context_section_description: 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) 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,
  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
caption: 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.
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