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

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

$$\begin{aligned} du=0, \end{aligned}$$

.