Properties of Differentiable Functions

Within Section 17, Servois explores the properties of what he calls differential functions, but what we would call differential operators. He begins with an arbitrary function \(z\) having only \(x\) and \(y\) as variables and then explores the varied states, differences, and differentials of \(z\), both total and partial, \[ Ez, \; \frac{E}{x}z, \; \frac{E}{y}z; \quad \Delta z, \; \frac{\Delta}{x}z, \; \frac{\Delta}{y}z; \quad {\mbox d} z, \; \frac{{\mbox d}}{x}z, \; \frac{{\mbox d}}{y}z.\] Servois had defined the partial varied state and partial difference in his equation (5) of Section 1. In this case, \(z = \varphi (x, y)\) and we have \[ \frac{E}{x}z = \varphi (x + \alpha, y) \quad \mbox{and} \quad \frac{E}{y}z = \varphi (x, y + \beta).\] The partial differences and differentials are defined from the partial varied states analogously with the total difference and total differential. Thus, all the results in this section are derived from the properties of the varied states. The varied states are distributive and all powers of the varied states are commutative with constant factors. Since differences are derived from varied states and differentials are found using differences, Servois concludes that these operators inherit the distributive and commutative properties of the varied states. Following this investigation, Servois makes the following observations about his class of differential functions:

1. All differential functions are distributive.
2. All differential functions, and their various orders, are commutative, both among themselves and with constant factors.

It is clear that integrals share these properties, since they represent the inverses of the differential operators. Servois had already defined the inverse \(\Sigma\) of the difference \(\Delta\) in Section 1, and he implicitly defines the integral \(\int\) here as the inverse of the differential operator \({\mbox d}\). Thus, \[\Sigma, \; \frac{\Sigma}{x}, \; \frac{\Sigma}{y}, \; \int,  \; \frac{\int}{x}, \;{\rm and}  \; \frac{\int}{y}\] can be added to the list of operators that are distributive and commutative with constant factors, as can their various orders. Servois observes that the results of this section are easily generalized for functions of more than two variables.

Servois concludes this section with the following algebraic theorem for differential functions: linear combinations of differential functions give rise to an infinite number of new differential functions, all of which are distributive and commutative among themselves and with constant factors. Therefore, Servois has shown that this large class of operators is linear, in the modern sense of items 1 and 2.