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This topic has appeared in the English Wikipedia rankings 1 time. It first appeared on 2026-03-15 and was most recently seen on 2026-03-15.
The triple product rule, known variously as the cyclic chain rule, cyclic relation, cyclical rule, Euler's chain rule, or the reciprocity theorem, 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
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