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

  • Dennis M. CatesEmail author
Chapter

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.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Sun CityUSA

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