Advertisement

USE OF DIFFERENTIALS OF VARIOUS ORDERS IN THE STUDY OF MAXIMA AND MINIMA OF FUNCTIONS OF SEVERAL VARIABLES.

  • Dennis M. CatesEmail author
Chapter

Abstract

Let \(u=f(x, y, z, \dots )\) be a function of the independent variables \( x, y, z, \dots , \) and set, as in the tenth lecture,
$$\begin{aligned} f(x+\alpha dx, y+\alpha dy, z+\alpha dz, \dots )=F(\alpha ). \end{aligned}$$
So that the value of u relative to certain particular values of \( x, y, z, \dots \) is either a maximum or a minimum, it will be necessary and sufficient that the corresponding value of \(F(\alpha )\) always becomes a maximum or a minimum, by virtue of the assumption \(\alpha =0. \ \) We conclude (see the tenth lecture) that the systems of values of \( x, y, z, \dots , \) which, without rendering discontinuous one of the two functions, u and du,  generates for the first, a maxima or a minima, and necessarily satisfies, regardless of \( dx, dy, dz, \dots , \) the equation
$$\begin{aligned} du=0, \end{aligned}$$
.

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Sun CityUSA

Personalised recommendations