This chapter introduces the Caratheodory formulation of the second law, which is that entropy exists as a state variable if there are states inaccessible from a given state by adiabatic paths. The Caratheodory formulation is equivalent to the Joule-Kelvin statement of the second law. A scalar field is a function that assigns a number to every point of its domain. In what follows people consider the conditions under which vector fields possess integrating factors. It turns out that Pfaffians in some variables are fundamentally different from those with three or more variables. A Pfaffian in some variables always possesses an integrating factor. The second law accomplishes the same tasks: Entropy and absolute temperature are consequences of the second law. Is the Caratheodory principle equivalent to the Kelvin and Rudolf Clausius statements of the second law, or is it implied by them. It is in the form of an impossibility statement; that much it shares with the other laws of thermodynamics.