E.1 Differentials

One of the peculiarities of the literature of thermodynamics is its widespread use of differentials such as dx. Differentials are infinitesimal quantities and are regarded by most mathematicians, and some physicists, ^[1] as very dubious entities, since it is difficult to define them a logically precise way. If, for example, one defines an infinitesimal as a quantity smaller than any nonzero number, it has to be zero, and a ratio of two differentials, such as , becomes meaningless. However, in this book we define dx as a change in x so small that quantities of the order of (dx) ² can be neglected. Such a quantity can, with caution, be treated as an ordinary variable so long as one does not change its order of magnitude; for example, is meaningless, since the square root of a very small number is much larger than that number. One way to avoid the use of differentials is to follow Bohren and Albrecht ^[1] in recognizing that any change must take some time, so that instead of writing dx, one writes , which can be defined rigorously in terms of a limit (see any textbook of analysis). This is logical, but it complicates the notation and introduces time as a variable where it is not relevant; we are not usually interested in the rate of change, but only in the fact that there is a change. However, anyone who is bothered by the presence of differentials...