METHODS THAT WORK TO SIMPLIFY THE STUDY OF TOTAL DIFFERENTIALS FOR FUNCTIONS OF SEVERAL INDEPENDENT VARIABLES. SYMBOLIC VALUES OF THESE DIFFERENTIALS.

Abstract

Let $u=f(x, y, z, \dots )$ always be a function of several independent variables $x, y, z, \dots ;$ and, denote by

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

its first-order partial derivatives relative to x,  to y,  to z,  $\dots . \$ If we make, as in the eighth lecture,

$$\begin{aligned} F(\alpha )=f(x+\alpha dx, y+\alpha dy, z+\alpha dz, \dots ), \end{aligned}$$

then, differentiate the two members of equation (1) with respect to the variable $\alpha ,$ we will find

$$\begin{aligned} \left\{ \begin{aligned} \ F^{\prime }(\alpha )&= \varphi (x+\alpha dx, y+\alpha dy, z+\alpha dz, \dots ) dx \\&\quad +\chi (x+\alpha dx, y+\alpha dy, z+\alpha dz, \dots ) dy \\&\quad +\psi (x+\alpha dx, y+\alpha dy, z+\alpha dz, \dots ) dz \\&\quad + \cdots . \end{aligned} \right. \end{aligned}$$

If, in this last formula, we set $\alpha =0,$ we will obtain the following

$$\begin{aligned} \left\{ \begin{aligned} \ F^{\prime }(0)=\varphi (x, y, z, \dots ) dx&+\chi (x, y, z, \dots ) dy \\&+\psi (x, y, z, \dots ) dz + \cdots =du, \end{aligned} \right. \end{aligned}$$

which is in accordance with equation (16) of the eighth lecture.