Foundations of Differential Calculus

ISBN-13: 9780387985343 / Angielski / Twarda / 2000 / 194 str.

Leonhard Euler;Euler; John D. Blanton

Opis

What differential calculus, and, in general, analysis ofthe infinite, might be can hardly be explainedto those innocent ofany knowledge ofit. Nor can we here offer a definition at the beginning of this dissertation as is sometimes done in other disciplines. It is not that there is no clear definition of this calculus; rather, the fact is that in order to understand the definition there are concepts that must first be understood. Besides those ideas in common usage, there are also others from finite analysis that are much less common and are usually explained in the courseofthe development ofthe differential calculus. For this reason, it is not possible to understand a definition before its principles are sufficiently clearly seen. In the first place, this calculus is concerned with variable quantities. Although every quantity can naturally be increased or decreased without limit, still, since calculus is directed to a certain purpose, we think of some quantities as being constantly the same magnitude, while others change through all the .stages of increasing and decreasing. We note this distinc tion and call the former constant quantities and the latter variables. This characteristic difference is not required by the nature of things, but rather because of the special question addressed by the calculus."

Recenzje

"This instalment ... will be welcomed not only by historians but also by teachers anxious to change from the highfalutin talk of Cauchy-style limits all the time when explaining the calculus." (I.Grattan-Guinness, zbMATH 0949.01030, 2021)

Spis treści

On Finite Differences.- On the Use of Differences in the Theory of Series.- On the Infinite and the Infinitely Small.- On the Nature of Differentials of Each Order.- On the Differentiation of Algebraic Functions of One Variable.- On the Differentiation of Transcendental Functions.- On the Differentiation of Functions of Two or More Variables.- On the Higher Differentiation of Differential Formulas.- On Differential Equations.