About the book

This course is designed as a textbook for engineering students. It embraces the topics in mathematical analysis usually included into curricula of technical colleges. The course also contains some optional material which may be omitted in a first reading of the book; the corresponding items are marked with the asterisk.
There are a number of courses dealing with special divisions of mathematical analysis, such as equations of mathematical physics, functions of a complex argument and the like, and therefore, although these divisions are important for mathematical education of an engineer, they are not treated in this book. We also draw attention to the fact that only a few questions related to approximate calculations and programming (e.g. the applica­tion of the differential to approximate calculations, methods of approximate solution of equations, numerical integration and solution of differential equations, etc.) are discussed in this course. For a thorough study of this subject some other textbooks should be used.

In this course many examples are given which demonstrate the application of mathematical analysis to various divisions of mechanics and physics. The study of these examples is very impor­tant since the main interest of an engineer lies in solving concrete applied problems. At the end of each chapter we give a number of questions aimed at checking the understanding of the theore­tical material. In the presentation of the material the main emphasis has been laid upon practical aspects, and some purely mathematical facts are given without proof. On the other hand, in some cases detailed proofs of theorems are given, especially when this elucidates the meaning of the theorem and shows in which way it can be applied. Besides, the study of the proofs helps the student to acquire practice in logical argument and provides prerequisites for further mathematical self-education.

Contents

Preface 5
Introduction 15

1. The Subject of Mathematical Analysis 15
2. Variables and Functions 15
3. The Role of Mathematics and Mathematical Analysis in Natural Sciences and Engineering 16

CHAPTER I. FUNCTION 19

§ 1. Real Numbers 19
§ 2. The Concept of Function 26
§ 3. Characteristics of Behaviour of Functions. Some Important Examples 38
§ 4. Inverse Function. Power, Exponential and Logarithmic Function 51
§ 5. Trigonometric, Inverse Trigonometric, Hyperbolic and Inverse
Hyperbolic Functions. 60

CHAPTER II. LIMIT. CONTINUITY 72

§ 1. Limit. Infinitely Large Magnitudes 72
§ 2. Continuous Functions 98
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§ 3. Comparison of Infinitesimals. Comparison of Infinitely Large
Magnitudes 109

CHAPTER III. DERIVATIVE AND DIFFERENTIAL. DIFFERENTIAL CALCULUS 118

§ 1. Derivative 118
§ 2. Differentiating Functions 126
§ 3. Some Geometrical Problems. Graphical Differentiation 148
§ 4. Differential 155
§ 5. Derivatives and Differentials of Higher Orders 167

CHAPTER IV. APPLICATION OF DIFFERENTIAL CALCULUS TO INVESTIGATION OF BEHAVIOUR OF FUNCTIONS 175

§ 1. Theorems of Fermat, Rolle, Lagrange and Cauchy 175
§ 2. Investigating Functions wita the Aid of First and Second Derivatives 181
§ 3. L’Hospital’s Rule. General Scheme for Investigating Functions 204
§ 4. Curvature 219
§ 5. Space Curves. Vector Function of a Scalar Argument 225
§ 6. Complex Functions of a Real Argument 237
§ 7. Solution of Equations 245
QUESTIONS 255

CHAPTER V. INTEGRAL CALCULUS 258

§ 1. Indefinite Integral 258
§ 2. Definite Integral 291
§ 3. Methods of Evaluating Definite Integrals. 318
§ 4. Improper Integrals 331

CHAPTER VI. APPLICATION OF INTEGRAL CALCULUS 345

§ 1. Some Problems of Geometry and Statics 345
§ 2. General Scheme of the Application of the Integral 358

CHAPTER VII. FUNCTIONS OF SEVERAL VARIABLES AND THEIR DIFFERENTIATION 365

§ 1. Functions of Several Variables 365
§ 2. Derivatives and Differentials. Differential Calculus 376
§ 3. Applications of Differential Calculus to Geometry 409
§ 4. Extrema of Functions of Two Variables 414
§ 5. Scalar Field 427

CHAPTER VIII. DOUBLE AND TRIPLE INTEGRAL 437

§ 1. Double Integrals 437
§ 2. Triple Integrals 459
§ 3. Integrals Dependent on Parameters 471

CHAPTER IX. LINE INTEGRALS AND SURFACE INTEGRALS. FIELD THEORY 482

§ 1. Line Integrals 482
§ 2. Surface Integrals 515
§ 3. Field Theory 533
Questions 561

CHAPTER X. DIFFERENTIAL EQUATIONS 564

§ 1. Differential Equations of the First Order 564
§ 2. Differential Equations of the Second and Higher Orders 592
§ 3. Linear Differential Equations 603
§ 4. Systems of Differential Equations 635
QUESTIONS 656

CHAPTER XI. SERIES 659

§ 1. Numerical Series 659
§ 2. Functional Series 678
§ 3. Power Series 684
§ 4. Expanding Functions into Power Series 691
§ 5. Some Applications of Taylor’s Series 706
§ 6*. Some Further Topics in the Theory of Power Series 716
Questions 721

CHAPTER XII. FOURIER SERIES AND FOURIER INTEGRAL. 724

§ 1. Fourier Series 724
§ 2. Some Further Topics in the Theory of Fourier Series 746
§ 3. Fourier-Integral 753
Questions 761

Table of Integrals 762

Bibliography 768

Name Index 770

Subject Index 772