Differential and Integral Calculus - Vol 1 - Richard Courant - Wiley (1988)

Differential and Integral Calculus - Vol 1 - Richard Courant - Wiley (1988)

(Parte 1 de 8)

Differential and Integral Calculus by R.Courant

Volume 1 I. Introduction

I. The Fundamental Ideas of the Integral and Differential Calculus

I. Differentiation and Integration of the Elementary Functions

IV. Further development of the Differential Calculus

V. Applications

VI. Taylor's Theorem and the Approximate Expressions of Functions by Polynomials

VII. Numerical Methods

VIII. Infinite Series and Other Limiting Processes

IX. Fourier Series

X. A Sketch of the Theory of Functions of Several Variables XI. The Differential Equations for the Simplest Types of Vibrations

Summary of Important Theorems and Formulae

Miscellaneous Exercises

Answers and Hints to Exercises Answers and Hints to Miscellaneous Exercises

Index

Differential and Integral Calculus

Chapter I: Introduction Contents

1.1 The Continuum of Numbers 1.6.6 The number π as a limit

1.1.1 The System of Rational Numbers and the Need for its Extension 1.7 The Concept of Limit where the V

1.1.2 Real Numbers and Infinite Domains 1.8 The Concept of Continuity

1.1.3 Expression of Numbers in Scales other than that of 10 1.8.1 Definitions

1.1.4 Inequalities 1.8.2 Points of Discontinuity

1.5 Schwarz's Inequality Exercises 1.1 1.8.3 Theorems on Continuous Functions

2. The Concept of Function Appendix I to Chapter I

1.2.1. Examples A1.1 The Principle of the Point of Accumulation and its Applications

1.2.2 Formulation of the Concept of Function A1.1.1 The Principle of the Point of Accumulation

1.2.3. Graphical Representation. Continuity. Monotonic Function A1.1.2. Limits of Sequences

1.2.4 Inverse Functions A1.1.3 Proof of Cauchy's Convergence Test

1.3 More Detailed Study of the Elementary Functions A1.1.4 The Existence of Limits of Bounded Monotonic Sequences:

1.3.1 The Rational Functions A1.1.5 Upper and Lower Points of Accumulation; Upper and Lower Bounds of a Set of Numberse of Limits of Bounded Monotonic

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1.3.2 The Algebraic Functions A1.2 Theorems on Continuous Functions

1.3.3 The Trigonometric Functions A1.2.1. Greatest and Least Values of Continuous functions

1.3.4 The Exponential Function and the Logarithm A1.2.2 The Uniformity of Continuity

1.4 Functions of an Integral variable. Sequences of Numbers

A1.2.3 The Intermediate Value Theorem

1. 5 The Concept of the Limit of a Sequence

1.6 Further Discussion of the Concept of Limit A1.2.5 Further Theorems on Continuous Functions

1.6.1 First Definition of Convergence A1.3 Some Remarks on the Elementary Functions

1.6.2 Second (Intrinsic) Definition of Convergence: Appendix I to Chapter I

1.6.3 Monotonic Sequence A2.1 Polar Co-ordinates

1.6.4 Operations with Limits A2.2. Remarks on Complex Numbers

1.6.5 The Number e

Chapter I.

Introduction

The differential and integral calculus is based on two concepts of outstanding importance, apart from the concept of number, namely, the concepts of function and limit. While these concepts can be recognized here and there even in the mathematics of the ancients, it is only in modern mathematics that their essential character and significance have been fully clarified. We shall attempt here to explain these concepts as simply and clearly as possible.

1.1 The Continuum of Numbers

We shall consider the numbers and start with the natural numbers 1, 2, 3, · as given as well as the rules

(a + b) + c = a + (b + c) - associative law of addition, a + b = b + a - commutative law of addition, (ab)c = a(bc) - associative law of multiplication, ab = ba - commutative law of multiplication, a(b + c) = ab + ac - distributive law of multiplication.

by which we calculate with them; we shall only briefly recall the way in which the concept of the positive integers (the natural numbers) has had to be extended.

(Parte 1 de 8)

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