THE DIFFERENTIAL OF THE SUM OF SEVERAL FUNCTIONS IS THE SUM OF THEIR DIFFERENTIALS. CONSEQUENCES OF THIS PRINCIPLE. DIFFERENTIALS OF IMAGINARY FUNCTIONS

Dennis M. Cates ()

Chapter Chapter 5 in Cauchy's Calcul Infinitésimal, 2019, pp 21-25 from Springer

Abstract: Abstract In previous lectures, we have shown how we form the derivatives and the differentials of functions of a single variable. We now add new developments to the study that we have made to this subject. Let x always be the independent variable and $$ \varDelta x=\alpha h=\alpha dx $$ an infinitely small increment attributed to this variable. If we denote by $$ s, u, $$ $$ v, w, \dots $$ several functions of x, and by $$ \varDelta s, \varDelta u, $$ $$ \varDelta v, \varDelta w, \dots $$ the simultaneous increments that they receive while we allow x to grow by $$ \varDelta x, $$ the differentials $$ ds, du, $$ $$ dv, dw, \dots $$ will be, according to their own definitions, respectively, equal to the limits of the ratios $$\begin{aligned} \frac{\varDelta s}{\alpha }, \ \ \frac{\varDelta u}{\alpha }, \ \ \frac{\varDelta v}{\alpha }, \ \ \frac{\varDelta w}{\alpha }, \ \ \dots . \end{aligned}$$

Date: 2019
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DOI: 10.1007/978-3-030-11036-9_5

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