Mathematical interlude Exact and inexact differentials

An exact differential integrates to a finite difference, Jf dU = U2 — U, which is independent of the path of integration. An inexact differential integrates to a total quantity, li 6 = 6 which depends on the path of integration. The cyclic integral of an exact differential is zero for any cycle, Eq. (7.7). The cyclic integral of an inexact differential is usually not zero. 

Note that the symbolism AQ and A VF is meaningless. If A VF meant anything, it would mean W2 — Wi, but the system in either the initial state or the final state does not have any work or W2, nor does it have any heat or Q2- Work and heat appear during a change in state they are not properties of the state, but properties of the path. 

Properties of the state of a system, such as T, p, V, U, have differentials that are exact. Differentials of properties of the path, such as Q and W, are inexact. For more properties of exact and inexact differentials see Section 9.6. 

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