DIFFERENTIALS OF ANY FUNCTION OF SEVERAL VARIABLES EACH OF WHICH IS IN ITS TURN A LINEAR FUNCTION OF OTHER SUPPOSED INDEPENDENT VARIABLES. DECOMPOSITION OF ENTIRE FUNCTIONS INTO REAL FACTORS OF FIRST OR OF SECOND DEGREE

Dennis M. Cates ()

Chapter Chapter 18 in Cauchy's Calcul Infinitésimal, 2019, pp 91-96 from Springer

Abstract: Abstract Let $$ a, b, c, \dots , k $$ be constant quantities, and let $$\begin{aligned} u=ax+by+cz+\cdots +k \end{aligned}$$ be a linear function of the independent variables $$ x, y, z, \dots . \ $$ The differential $$\begin{aligned} du=a dx+b dy+c dz+\cdots \end{aligned}$$ will itself be a constant quantity, and as a result, the differentials $$ d^2u, $$ $$ d^3u, $$ $$\dots $$ will all be reduced to zero. We immediately conclude from this remark that the successive differentials of the functions $$\begin{aligned} f(u), \ \ \ f(u, v), \ \ \ f(u, v, w, \dots ), \ \ \ \dots \end{aligned}$$

Date: 2019
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DOI: 10.1007/978-3-030-11036-9_18

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