In mathematics, a cycle integral, mathematical operator ∮ (link), or the integration sign with a circle in it:
sometimes called the closed path integral, or circle integral, and refers to an "integration is done around a closed path" of a path-dependent function, i.e. state variable, as compared to an path-independent function, such as heat or work, specifically with reference to the a body returning to its "ideal original condition", as defined by the reversible cycle (reversible process), or "non-ideal original condition", as defined by the irreversible cycle (irreversible process).
History
The mathematics of the cycle integral stem from German physicist Rudolf Clausius' 1858 article “On the Treatment of Differential Equations which are Not Directly Integrable”, published in Dingler’s Polytechnisches Journal, which was then expanded on as the "mathematical introduction" section to the first (1865) and second (1875) editions of The Mechanical Theory of Heat. [1] The logic of this derivation, however, seems to predate Clausius and needs to be tracked down.
References
1. (a) Clausius, Rudolf. (1858). “On the Treatment of Differential Equations which are not Directly Integrable.” Dingler’s Polytechnisches Journal, vol. cl. (pg. 29).
(b) Clausius, Rudolf. (1865). The Mechanical Theory of Heat (section: On the Treatment of Differential Equations which are not Directly Integrable, pgs. 1-13). London: Macmillan & Co.
(c) Clausius, Rudolf. (1875). The Mechanical Theory of Heat (section: Mathematical Introduction: on Mechanical Work, on Energy, and on the Treatment of Non-Integrable Differential Equations, pgs. 1-20). London: Macmillan & Co.
