DIFFERENTIALS OF FUNCTIONS OF SEVERAL VARIABLES. PARTIAL DERIVATIVES AND PARTIAL DIFFERENTIALS.

Abstract

Let $u=f(x, y, z, \dots )$ be a function of several independent variables $x, y,$ $z, \dots . \$ We denote by i an infinitely small quantity, and by

$$\begin{aligned}&\varphi (x, y, z, \dots ), \\&\chi (x, y, z, \dots ), \\&\psi (x, y, z, \dots ), \\&\ \ \dots \dots \dots \dots \end{aligned}$$

the limits toward which the ratios

$$\begin{aligned}&\frac{f(x+i, y, z, \dots )-f(x, y, z, \dots )}{i}, \\&\frac{f(x, y+i, z, \dots )-f(x, y, z, \dots )}{i}, \\&\frac{f(x, y, z+i, \dots )-f(x, y, z, \dots )}{i}, \\&\ \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \end{aligned}$$

converge, while i indefinitely approaches zero.