Any reversible process can be described as a sum of many infinitesimally small Carnot cycles, so $\oint {dS} = \oint {\frac{{dQ}}{T}} = 0 % MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHb5MDXbpmVaibaieYlf9irVe % eu0dXdh9vqqj-hEeeu0xXdbba9frFj0-OqFfea0dXdd9vqaq-JfrVk % FHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaci % aacaqabeaadaqaaqaafaGcqaaaaaaaaaWdbeaadaWdfaqaaiaadsga % caWGtbaaleqabeqdcqWIr4E0cqGHRiI8aOGaeyypa0Zaa8qbaeaada % WcaaqaaiaadsgacaWGrbaabaGaamivaaaaaSqabeqaniablgH7rlab % gUIiYdGccqGH9aqpcaaIWaaaaa!5091! $ holds. It means the integral is independent of path it takes so the entropy S is a state variable. Such a path-independence is only true so reversible process, in strickly speaking. Then...is the entropy S not state variable for irreversible process?