Foundations of Differential Calculus

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Springer Science & Business Media, May 23, 2000 - Mathematics - 194 pages
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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.
 

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Contents

On Finite Differences
1
On the Use of Differences in the Theory of Series
25
On the Infinite and the Infinitely Small
47
On the Nature of Differentials of Each Order
63
On the Differentiation of Algebraic Functions of One Variable
77
On the Differentiation of Transcendental Functions
99
On the Differentiation of Functions of Two or More Variables
123
On the Higher Differentiation of Differential Formulas
141
On Differential Equations
167
Index
193
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About the author (2000)

Leonhard Euler was one of the most prolific mathematicians of all time, amassing nearly 900 publications over the course of his lifetime. Born in Basel, Switzerland, Euler spent substantial amounts of time promoting mathematics at the courts of Berlin and St. Petersburg. Euler was adept at pure and applied mathematics. His textbooks on algebra and calculus became classics and for generations remained standard introductions to both subjects. He also made seminal advances in the theory of differential equations, number theory, mechanics, astronomy, hydraulics, and the calculus of variations. In 1738, Euler lost vision in one eye. In time, he became totally blind but continued to write. During his life, Euler published more than 800 books, most of them in Latin. Euler died in 1783.

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