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Über dieses Buch

From the Preface: (...) The book is addressed to students on various levels, to mathematicians, scientists, engineers. It does not pretend to make the subject easy by glossing over difficulties, but rather tries to help the genuinely interested reader by throwing light on the interconnections and purposes of the whole. Instead of obstructing the access to the wealth of facts by lengthy discussions of a fundamental nature we have sometimes postponed such discussions to appendices in the various chapters. Numerous examples and problems are given at the end of various chapters. Some are challenging, some are even difficult; most of them supplement the material in the text. In an additional pamphlet more problems and exercises of a routine character will be collected, and moreover, answers or hints for the solutions will be given. This first volume of concerned primarily with functions of a single variable, whereas the second volume will discuss the more ramified theories of calculus (...).

Inhaltsverzeichnis

1. Introduction

Abstract
Since antiquity the intuitive notions of continuous change, growth, and motion, have challenged scientific minds. Yet, the way to the understanding of continuous variation was opened only in the seventeenth century when modern science emerged and rapidly developed in close conjunction with integral and differential calculus, briefly called calculus, and mathematical analysis.
Richard Courant, Fritz John

2. The Fundamental Ideas of the Integral and Differential Calculus

Abstract
The fundamental limiting processes of calculus are integration and differentiation. Isolated instances of these processes of calculus were considered even in antiquity (culminating in the work of Archimedes), and with increasing frequency in the sixteenth and seventeenth centuries. However, the systematic development of calculus, started only in the seventeenth century, is usually credited to the two great pioneers of science, Newton and Leibnitz. The key to this systematic development is the insight that the two processes of differentiation and integration, which had been treated separately, are intimately related by being reciprocal to each other.1
Richard Courant, Fritz John

3. The Techniques of Calculus

Abstract
Although problems of integration are usually of greater importance than those of differentiation, the latter offer less formal difficulty than the former. Therefore it is a natural procedure first to master the art of differentiating the widest possible classes of functions; then by the fundamental theorem (Section 2.9) the results of differentiation are available for evaluating integrals. In the following sections we shall pursue such applications of the fundamental theorem. To a certain extent we shall make a fresh start and develop techniques of integration systematically on the basis of certain general rules for differentiation.
Richard Courant, Fritz John

4. Applications in Physics and Geometry

Abstract
The representation of a curve by an equation y = f(x) imposes a serious geometrical restriction: A curve so represented must not be intersected at more than one point by any parallel to the y-axis. Usually, this restriction can be overcome by decomposing the curve into portions each representable in the form y = f(x). Thus a circle of radius a about the origin is given by the two functions $$y = \sqrt {{a^2} - {x^2}}$$ and $$y = - \sqrt {{a^2} - {x^2}}$$ defined for −axa. However, for as simple a curve as a parallel to the y-axis this device does not work.
Richard Courant, Fritz John

5. Taylor’s Expansion

Abstract
It was a great triumph in the early years of Calculus when Newton and others discovered that many known functions could be expressed as “polynomials of infinite order” or “power series,” with coefficients formed by elegant transparent laws. The geometrical series for 1/(1 − x) or 1/(1 + x2)
$$\frac{1}{{1\; - \;x}} = 1 + x + {x^2} \cdots + {x^n} + \cdots$$
(1)
$$\frac{1}{{1 + {x^2}}} = 1 - {x^2} + {x^4} - {x^6} + \cdots + {( - 1)^n}{x^{2n}} + \cdots$$
(1a)
valid for the open interval |x| < 1, are prototypes (see Chapter 1, p. 67).
Richard Courant, Fritz John

6. Numerical Methods

Abstract
The task of solving an analytical problem always remains uncompleted. The proof of the existence and of some basic properties of the solution is usually considered satisfactory, but relevant questions always remain to be answered. Thus, when the solution is defined by a limit process, for example by an integral, the problem arises of actually finding approximations to this limit and of estimating the accuracy of these approximations. Not only are such questions of basic importance theoretically but they are also inevitable, if we wish to apply analysis to the description and control of natural phenomena which in principle can be described only in an approximate manner.
Richard Courant, Fritz John

7. Infinite Sums and Products

Abstract
The geometric series, Taylor’s series, and a number of examples previously discussed in this book, suggest that we may well study those limiting processes of analysis which involve the summation of infinite series from a more general point of view. In principle, any limiting value
$$S = \mathop {\lim }\limits_{n \to \infty } {s_n}$$
can be written as an infinite series; we need only put $${a_n} = {s_n} - {s_{n - 1}}$$ for n > 1 and al = sl to obtain
$${s_n}={a_1}+{a_2}+\cdots +{a_n},$$
and the value S thus appears as the limit of sn, the sum of n terms, as n increases. We express this fact by saying that S is the “sum of the infinite series”
$${a_1} + {a_2} + {a_3} + \cdots$$
.
Richard Courant, Fritz John

8. Trigonometric Series

Abstract
The functions represented by power series, or as Lagrange called them, the “analytic functions,” play indeed a central role in analysis. But the class of analytic functions is too restricted in many instances. It was therefore an event of major importance for all of mathematics and for a great variety of applications when Fourier in his “Théorie analytique de la chaleur”1 observed and illustrated by many examples the fact that convergent trigonometric series of the form
$$f(x)=\frac{{{a_0}}}{2}+\sum\limits_{v = 1}^\infty {({a_v}\cos{\text{}}vx+{b_v}\sin{\text{}}vx)}$$
(1)
with constant coefficients a v , b v are capable of representing a wide class of “arbitrary” functions f(x), a class which includes essentially every function of specific interest, whether defined geometrically by mechanical means, or in any other way: even functions possessing jump discontinuities, or obeying different laws of formation in different intervals, can thus be expressed.
Richard Courant, Fritz John

9. Differential Equations for the Simplest Types of Vibration

Abstract
On several previous occasions we have met with differential equations, that is, equations from which an unknown function is to be determined and which involve not only this function itself but also its derivatives.
Richard Courant, Fritz John

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