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

Dennis M. Cates ()

Chapter Chapter 16 in Cauchy's Calcul Infinitésimal, 2019, pp 81-84 from Springer

Abstract: 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 $$\begin{aligned} du=0, \end{aligned}$$ .

Date: 2019
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DOI: 10.1007/978-3-030-11036-9_16

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