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2000 | Buch

On Finite Differences Kapitel lesen Erstes Kapitel lesen

Foundations of Differential Calculus

verfasst von:  Euler

Verlag: Springer New York

Enthalten in: Professional Book Archive

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter
1. On Finite Differences
Abstract
From what we have said in a previous book1 about variables and functions, it should be clear enough that as a variable changes, the values of all functions dependent on that variable also change. Thus if a variable quantity x changes by an increment ω, instead of x we write x + ω.
Euler
2. On the Use of Differences in the Theory of Series
Abstract
It is well known that the nature of series can be very well illustrated from first principles through differences. Indeed, arithmetic progressions, which are ordinarily considered first, have this particular property, that their first differences are equal to each other. From this it follows that their second differences and all higher differences will vanish.
Euler
3. On the Infinite and the Infinitely Small
Abstract
Since every quantity, no matter how large, can always be increased, and there is no obstacle to adding to a given quantity another like quantity, it follows that every quantity can be increased without limit. Furthermore, there is no quantity so large that a larger one cannot be conceived, and so there is no doubt that every quantity can be increased to infinity. If there is someone who would deny this, he would have to give some quantity that cannot be increased, and so he needs to give a quantity to which nothing can be added. This is absurd, and even the idea of quantity rules out this possibility. He must necessarily concede that every quantity can always be increased without limit, that is, it can be increased to infinity.
Euler
4. On the Nature of Differentials of Each Order
Abstract
In the first chapter we saw that if the variable quantity x received an increment equal to ω, then from this each function of x obtained an increment that can be expressed as Pω + Qω 2 + Rω 3 +..., and this expression may be finite or it may go to infinity.
Euler
5. On the Differentiation of Algebraic Functions of One Variable
Abstract
Since the differential of the variable x is equal to dx, when x is incremented, x becomes equal to x + dx. Hence, if y is some function of x, and if we substitute x + dx for x, we obtain y I. The difference y I - y. gives the differential of y.
Euler
6. On the Differentiation of Transcendental Functions
Abstract
Besides the infinite class of transcendental, or nonalgebraic, quantities that integral calculus supplies in abundance, in Introduction to Analysis of the Infinite we were able to gain some knowledge of more usual quantities of this kind, namely, logarithms and circular arcs. In that work we explained the nature of these quantities so clearly that they could be used in calculation with almost the same facility as algebraic quantities.
Euler
7. On the Differentiation of Functions of Two or More Variables
Abstract
If two or more variable quantities x, y, z are independent of each other, it can happen that while one of the variables increases or decreases, the other variables remain constant. Since we have supposed that there is no connection between the variables, a change in one does not affect the others. Neither do the differentials of the quantities y and z depend on the differential of x, with the result that when x is increased by its differential dx, the quantities y and z can either remain the same, or they can change in any desired way. Hence, if the differential of x is dx, the differentials of the remaining quantities, dy and dz, remain indeterminate and by our arbitrary choice will be presumed to be either practically nothing or infinitely small when compared to dx.
Euler
8. On the Higher Differentiation of Differential Formulas
Abstract
If there is a single variable and its differential is held constant, we have already given the method for finding differentials of any order. That is, if the differential of any function is differentiated again, we obtain its second differential. If this is again differentiated, we get the third differential, and so forth. This same rule holds whether the function involves several variables or only one, whose first differential is not kept constant.
Euler
9. On Differential Equations
Abstract
In this chapter we principally set forth an explanation of the differentiation of those functions of x that are not defined explicitly, but implicitly by means of the relationship of x to the function y. Once this is accomplished, we consider the nature of differential equations in general, and we show how they arise from finite equations.
Euler
Backmatter
Metadaten
Titel
Foundations of Differential Calculus
verfasst von
Euler
Copyright-Jahr
2000
Verlag
Springer New York
Electronic ISBN
978-0-387-22645-3
Print ISBN
978-0-387-98534-3
DOI
https://doi.org/10.1007/b97699