In this post, we will see the book *Introductory Mathematics for Engineers: Lectures in Higher Mathematics *by A. D. My*š*kis. The book is around 800 pages and is very exhaustive in the number of topics it deals in. Starting from functions and their graphs, it traverses the mathematical landscape of plane and solid analytic geometry, limits and continuity, matrices and systems of linear equations, differential and integral calculus, definite, indefinite and multiple integrals, partial derivatives, differential equations, vectors, linear algebra, complex numbers, functions of several variables, series, probability etc.

About The Book

The present book is based on lectures given by the author over a number of years to students of various eng1!1eermg and physics. The book includes some optional can be skipped for the first reading. The corresponding Items m the table of contents are marked by an asterisk.

…

The book is composed in such a way that it is possible to use it both for studying in a college under the guidance of a teacher and for self-education. The subject matter of the book is divided into small sections so that the reader could study the material in suitable order and to any extent depending on the profession and the needs of the reader. It is also intended that the book can be used by students taking a correspondence course and by the readers who have some prerequisites in higher mathematics and want to perfect their knowledge by reading some chapters of the book.

…

The book can be of use to readers of various professions dealing with applications of mathematics in their work. Modern applied mathematics of many important special divisions which are not included m this book. The author intends to write another book devoted to some supplementary topics such as the theory of functions of a complex argument, variational calculus, mathematical physics, some special questions of the theory of ordinary differential equations and so on.

The book has interesting ways to treat affine mappings (pages 344-345) and non-linear mappings (pages 358-359).

The book was translated from the Russian by V. M. Volosov and was first published by Mir in 1972.

PDF | OCR | Bookmarked | Cover | 787 pages

The Internet Archive link

Note: quite a few pages are missing from the scan:

56-57 70-71 210-211 240-241 312-313 315 320-321 337-338 338-339-340 418-419 464-465 759-760 764-765

We would be grateful if anyone points to a copy with the missing pages.

Credits to the original scanner. The original scan was not clean or bookmarked. We cleaned, OCRed and bookmarked the original scan.

Contents

Front Cover

Title Page

Preface 5

Contents

Introduction 19

1. The Subject of Mathematics 19

2. The Importance of Mathematics and Mathematical Education 20

3. Abstractness 20

4. Characteristic Features of Higher Mathematics 22

5. Mathematics in the Soviet Union 23

CHAPTER I. VARIABLES AND FUNCTIONS 25

§ 1. Quantities 25

1. Concept of a Quantity 25

2. Dimensions of Quantities 25

3. Constants and Variables 26

4. Number Scale. Slide Rule 27

5. Characteristics of Variables 29

§ 2. Approximate Values of Quantities 32

6. The Notion of an Approximate Value 32

7. Errors 32

8. Writing Approximate Numbers 33

9. Addition and Subtraction of Approximate Numbers 34

10. Multiplication and Division of Approximate Numbers Remarks 36

§ 3. Functions and Graphs 39

11. Functional Relation 39

12. Notation 40

13. Methods of Representing Functions 42

14. Graphs of Functions 45

15. The Domain of Definition of a Function 47

16. Characteristics of Behaviour of Functions 48

17. Algebraic Classification of Functions 51

18. Elementary Functions 53

19. Transforming Graphs 54

20. Implicit Functions 56

21. Inverse Functions 58

§ 4. Review of Basic Functions 60

22. Linear Function 60

23. Quadratic Function 62

24. Power Function 63

25. Linear-Fractional Function 66

26. Logarithmic Function 68

27. Exponential Function 69

28. Hyperbolic Functions 70

29. Trigonometric Functions 72

30. Empirical Formulas 75

CHAPTER II. PLANE ANALYTIC GEOMETRY 78

§ 1. Plane Coordinates 78

1. Cartesian Coordinates 78

2. Some Simple Problems Concerning Cartesian Coordinates 79

3. Polar Coordinates 81

§ 2. Curves in Plane 82

4. Equation of a Curve in Cartesian Coordinates 82

5. Equation of a Curve in Polar Coordinates 84

6. Parametric Representation of Curves and Functions 87

7. Algebraic Curves 90

8. Singular Cases 92

§ 3. First-Order and Second-Order Algebraic Curves 94

9. Curves of the First Order 94

10. Ellipse 96

11. Hyperbola 99

12. Relationship Between Ellipse, Hyperbola and Parabola 102

13. General Equation of a Curve of the Second Order 105

CHAPTER III. LIMIT. CONTINUITY 109

§ 1. Infinitesimal and Infinitely Large Variables 109

1. Infinitesimal Variables 109

2. Properties of Infinitesimals 111

3. Infinitely Large Variables 112

§ 2. Limits 113

4. Definition 113

5. Properties of Limits 115

6. Sum of a Numerical Series 117

§ 3. Comparison of Variables 121

7. Comparison of Infinitesimals 121

8. Properties of Equivalent Infinitesimals 122

9. Important Examples 122

10. Orders of Smallness 124

11. Comparison of Infinitely Large Variables 125

§ 4. Continuous and Discontinuous Functions 125

12. Definition of a Continuous Function 125

13. Points of Discontinuity 126

14. Properties of Continuous Functions 129

15. Some Applications 131

CHAPTER IV. DERIVATIVES, DIFFERENTIALS, INVESTIGATION OF THE BEHAVIOUR OF FUNCTIONS 134

§ 1. Derivative 134

1. Some Problems Leading to the Concept of a Derivative 134

2. Definition of Derivative 136

3. Geometrical Meaning of Derivative 137

4. Basic Properties of Derivatives 139

5. Derivatives of Basic Elementary Functions 142

6. Determining Tangent in Polar Coordinates 146

§ 2. Differential 148

7. Physical Examples 148

8. Definition of Differential and Its Connection with Increment 149

9. Properties of Differential 152

10. Application of Differentials to Approximate Calculations 153

§ 3. Derivatives and Differentials of Higher Orders 155

11. Derivatives of Higher Orders 155

12. Higher-Order Differentials 156

§ 4. L’Hospital’s Rule 158

13. Indeterminate Forms of the Type $\dfrac{0}{0}$ 158

14. Indeterminate Forms of tl1e Type $\dfrac{\infty}{\infty}$ 160

§ 5. Taylor’s Formula and Series 161

15. Taylor’s Formula 161

16. Taylor’s Series 163

§ 6. Intervals of Monotonicity. Exrtremum 165

17. Sign of Derivative 165

18. Points of Extremum 166

19. The Greatest and the Least Values of a Function 168

§ 7. Constructing Graphs of Functions 173

20. Intervals of Convexity of a Graph and Points of Inflection 173

21. Asymptotes of a Graph 174

22. General Scheme for Investigating a Function and Constructing Its Graph 175

CHAPTER V. APPROXIMATING ROOTS OF EQUATIONS. INTERPOLATION 179

§ 1. Approximating Roots of Equations 179

1. Introduction 179

2. Cut-and-Try Method. Method of Chords. Method of Tangents 181

3. Iterative Method 185

4. Formula of Finite Increments 187

5*. Small Parameter Method 189

§ 2. Interpolation 191

6. Lagrange’s Interpolation Formula 191

7. Finite Differences and Their Connection with Derivatives 192

8. Newton’s Interpolation Formulas 196

9. Numerical Differentiation 198

CHAPTER VI. DETERMINANTS AND SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS 200

§ 1. Determinants 200

1. Definition 200

2. Properties 201

3. Expanding a Determinant in Minors of Its Row or Column 203

§ 2. Systems of Linear Algebraic Equations 206

4. Basic Case 206

5. Numerical Solution 208

6. Singular Case 209

CHAPTER VII. VECTORS 212

§ 1. Linear Operations on Vectors 212

1. Scalar and Vector Quantities 212

2. Addition of Vectors 213

3. Zero Vector and Subtraction of Vectors 215

4. Multiplying a Vector by a Scalar 215

5. Linear Combination of Vectors 216

§ 2. Scalar Product of Vectors 219

6. Projection of Vector on Axis 219

7. Scalar Product 220

8. Properties of Scalar Product 221

§ 3. Cartesian Coordinates in Space 222

9. Cartesian Coordinates in Space 222

10. Some Simple Problems Concerning Cartesian Coordinates 223

§ 4. Vector Product of Vectors 227

11. Orientation of Surface and Vector of an Area 227

12. Vector Product 228

13. Properties of Vector Products 230

14*. Pseudovectors 233

§ 5. Products of Three Vectors 235

15. Triple Scalar Product 235

16. Triple Vector Product 236

§ 6. Linear Spaces 237

17. Concept of Linear Space 237

18. Examples 239

19. Dimension of Linear Space 241

20. Concept of Euclidean Space 244

21. Orthogonality 245

§ 7. Vector Functions of Scalar Argument. Curvature 248

22. Vector Variables 248

23. Vector Functions of Scalar Argument 248

24. Some Notions Related to the Second Derivative 251

25. Osculating Circle 252

26. Evolute and Evolvent 255

CHAPTER VIII. COMPLEX NUMBERS AND FUNCTIONS 259

§ 1. Complex Numbers 259

1. Complex Plane 259

2. Algebraic Operations on Complex Numbers 261

3. Conjugate Complex Numbers 263

4. Euler’s Formula 264

5. Logarithms of Complex Numbers 266

§ 2. Complex Functions of a Real Argument 267

6. Definition and Properties 267

7*. Applications to Describing Oscillations 269

§ 3. The Concept of a Function of a Complex Variable 271

8. Factorization of a Polynomial 271

9*. Numerical Methods of Solving Algebraic Equations 273

10. Decomposition of a Rational Fraction into Partial Rational Fractions 277

11*. Some General Remarks on Functions of a Complex Variable 280

CHAPTER IX. FUNCTIONS OF SEVERAL VARIABLES 283

§ 1. Functions of Two Variables 283

1. Methods of Representing 283

2. Domain of Definition 286

3. Linear Function 287

4. Continuity and Discontinuity 288

5. Implicit Functions 291

§ 2. Functions of Arbitrary Number of Variables 291

6. Methods of Representing 291

7. Functions of Three Arguments 292

8. General Case 292

9. Concept of Field 293

§ 3. Partial Derivatives and Differentials of the First Order 294

10. Basic Definitions 294

11. Total Differential 296

12. Derivative of Composite Function 298

13. Derivative of Implicit Function 300

§ 4. Partial Derivatives and Differentials of Higher Orders 303

14. Definitions 303

15. Equality of Mixed Derivatives 304

16. Total Differentials of Higher Order 305

CHAPTER X. SOLID ANALYTIC GEOMETRY 307

§ 1. Space Coordinates 307

1. Coordinate Systems in Space 307

2*. Degrees of Freedom 309

4. Cylinders, Cones and Surfaces of Revolution 314

5. Curves In Space 316

6. Parametric Representation of Surfaces in Space. Parametric Representation of Functions of Several Variables 317

§ 3. Algebraic Surfaces of the First and of the Second Orders 319

7. Algebraic Surfaces of the First Order 319

8. Ellipsoids 322

9. Hyperboloids 324

10. Paraboloids 326

11. General Review of the Algebraic surfaces of the second order 327

CHAPTER XI. MATRICES AND THEIR APPLICATIONS 329

§ 1. Matrices 329

1. Definitions 329

2. Operations on Matrices 331

3. Inverse Matrix 333

4. Eigenvectors and Eigenvalues of a Matrix 335

5. The Rank of a Matrix 337

7. Transformation of the Matrix of a Linear Mapping When the Basis Is Changed 347

8. The Matrix of a Mapping Relative to the Basis Consisting of Its Eigenvectors 350

9. Transforming Cartesian Basis 352

10. Symmetric Matrices 353

§ 3. Quadratic Forms 355

11. Quadratic Forms 355

12. Simplification of Equations of Second-Order Curves and Surfaces 357

§ 4. Non-Linear Mappings 358

13*. General Notions 358

14*. Non Linear Mapping in the Small 360

15*. Functional Relation Between Functions 362

CHAPTER XII. APPLICATIONS OF PARTIAL DERIVATIVES 365

§ 1. Scalar Field 365

1. Directional Derivative. Gradient 365

2. Level Surfaces 368

3. Implicit Functions of Two Independent Variables 370

4. Plane Fields 371

5. Envelope of One-Parameter Family of Curves 372

§ 2. Extremum of a Function of Several Variables 374

6. Taylor’s Formula for a Function of Several Variables 374

7. Extremum 375

8. The Method of Least Squares 380

9*. Curvature of Surfaces 381

10. Conditional Extremum 384

11. Extremum with Unilateral Constraints 388

12*. Numerical Solution of Systems of Equations 390

CHAPTER XIII. INDEFINITE INTEGRAL 393

§ 1. Elementary Methods of Integration 393

1. Basic Definitions 393

2. The Simplest Integrals 394

3. The Simplest Properties of an Indefinite Integral 397

4. Integration by Parts 399

5. Integration by Change of Variable (by Substitution) 402

§ 2. Standard Methods of Integration 404

6. Integration of Rational Functions 405

7. Integration of Irrational Functions Involving Linear and Linear-Fractional Expressions 407

8. Integration of Irrational Expressions Containing Quadratic Trinomials 408

9. Integrals of Binomial Differentials 411

lO. Integration of Functions Rationally Involving Trigonometric Functions 412

11. General Remarks 415

CHAPTER XIV. DEFINITE INTEGRAL 417

§ 1. Definition and Basic Properties 417

1. Examples Lending to the Concept of Definite Integral 417

3. Relationship Between Definite Integral and Indefinite Integral 426

4. Basic Properties of Definite Integral 433

5. Integrating Inequalities 436

§ 2. Applications of Definite Integral 436

6. Two Schemes of Application 436

7. Differential Equations with Variables Separable 437

8. Computing Areas of Plane Geometric Figures 443

9. The Arc Length of a Curve 445

10. Computing Volumes of Solids 447

11. Computing Area of Surface of Revolution 448

§ 3. Numerical Integration 448

12. General Remarks 448

13. Formulas of Numerical Integration 450

§ 4. Improper Integrals 454

14. Integrals with Infinite Limits of Integration 455

15. Basic Properties of Integrals with Infinite Limits 464

16. Other Types of Improper Integral 468

17*. Gamma Function 468

18*. Beta Function 471

19*. Principal Value of Divergent Integral 473

§ 5. Integrals Dependent on Parameters 474

20*. Proper Integrals 474

21*· Improper Integrals 476

§ 6. Line Integrals of Integration 478

22. Line Integrals of the First Type 482

23. Line Integrals of the Second Type 484

24. Conditions for a Line Integral of the Second Type to Be Independent of the Path of Integration 488

§ 7. The Concept of Generalized Function 488

25*. Delta Function 488

26*. Application to Constructing Influence Function 492

27*. Other Generalized Functions 495

CHAPTER XV. DIFFERENTIAL EQUATIONS 497

§ 1. General Notions 497

1. Examples 497

2. Basic Definitions 498

§ 2. First-Order Differential Equations 500

3. Geometric Meaning 500

4. Integrable Types of Equations 503

5*. Equation for Exponential Function 506

6. Integrating Exact Differential Equations 509

7. Singular Points and Singular Solutions 512

8. Equations Not Solved for the Derivative 516

9. Method of Integration by Means of Differentiation 517

§ 3. Higher-Order Equations and Systems of Differential Equations 519

10. Higher-Order Differential Equations 519

11*. Connection Between Higher-Order Equations and Systems of First-Order Equations 521

12*. Geometric Interpretation of System of First-Order Equations 522

13*. First Integrals 526

§ 4. Linear Equations of General Form 528

14. Homogeneous Linear Equations 528

15. Non-Homogeneous Equations 530

16*. Boundary-Value Problems 535

§ 5. Linear Equations with Constant Coefficients 541

17. Homogeneous Equations 541

18. Non-Homogeneous Equations with Right-Hand Sides of Special Form 545

19. Euler’s Equations 548

20*. Operators and the Operator Method of Solving Differential Equations 549

§ 6. Systems of Linear Equations 553

21. Systems of Linear Equations 553

22*. Applications to Testing Lyapunov Stability of Equilibrium State 558

§ 7. Approximate and Numerical Methods of Solving Differential Equations 562

23. Iterative Method 562

24*. Application of Taylor’s Series 564

25. Application of Power Series with Undetermined coefficients 565

26*. Bessel’s Functions 566

27*. Small Parameter Method 569

28*. General Remarks on Dependence of Solutions on Parameters 572

29*. Methods of Minimizing Discrepancy 575

30*. Simplification Method 576

31. Euler’s Method 578

32. Runge-Kutta Method 580

33. Adams Method 582

34. Milne’s Method 583

CHAPTER XVI. Multiple Integrals 585

§ 1. Definition and Basic Properties of Multiple Integrals 585

1. Some Examples Leading to the Notion of a Multiple Integral 585

2. Definition of a Multiple Integral 586

3. Basic Properties of Multiple Integrals 587

4. Methods of Applying Multiple Integrals 589

5. Geometric Meaning of an Integral Over a Plane Region 591

§ 2. Two Types of Physical Quantities 592

6*. Basic Example. Mass and Its Density 592

7*. Quantities Distributed in Space 594

§ 3. Computing Multiple Integrals in Cartesian Coordinates 596

8. Integral Over Rectangle 596

9. Integral Over an Arbitrary Plane Region 599

10. Integral Over an Arbitrary Surface 602

11. Integral Over a Three-Dimensional Region 604

§ 4. Change of Variables in Multiple Integrals 605

12. Passing to Polar Coordinates in Plane 605

13. Passing to Cylindrical and Spherical Coordinates 606

14*. Curvilinear Coordinates in Plane 608

15*. Curvilinear Coordinates in Space 611

16*. Coordinates on a Surface 612

§ 5. Other Types of Multiple Integrals 615

17*. Improper Integrals 615

18*. Integrals Dependent on a Parameter 617

19*. Integrals with Respect to Measure. Generalized Functions 620

20*. Multiple Integrals of Higher Order 622

§ 6. Vector Field 626

21*. Vector Lines 626

22*. The Flux. of a Vector Through a Surface 627

23*. Divergence 629

24*. Expressing Divergence in Cartesian Coordinates 632

25. Line Integral and Circulation 634

26*. Rotation 634

27. Green’s Formula. Stokes’ Formula 638

28*. Expressing Differential Operations on Vector Fields in a Curvilinear Orthogonal Coordinate System 641

29*. General Formula for Transforming Integrals 642

CHAPTER XVII. SERIES 645

§ 1. Number Series 645

1. Positive Series 645

2. Series with Terms of Arbitrary Signs 650

3. Operations on Series 652

4*. Speed of Convergence of a Series 654

5. Series with Complex, Vector and Matrix Terms 658

6. Multiple Series 659

§ 2. Functional Series 661

7. Deviation of Functions 661

8. Convergence of a Functional Series 662

9. Properties of Functional Series 664

§ 3. Power Series 666

10. Interval of Convergence 666

11. Properties of Power Series 667

12. Algebraic Operations on Power Series 671

13. Power Series as a Taylor Series 675

14. Power Series with Complex Terms 676

15*. Bernoullian Numbers 677

16*. Applying Series to Solving Difference Equations 678

17*. Multiple Power Series 680

18*. Functions of Matrices 681

19*. Asymptotic Expansions 685

§ 4. Trigonometric Series 686

20. Orthogonality 686

21. Series in Orthogonal Functions 689

22. Fourier Series 690

23. Expanding a Periodic Function 695

24*. Example. Bessel’s Functions as Fourier Coefficients 697

25. Speed of Convergence of a Fourier Series 698

26. Fourier Series in Complex Form 702

27*. Parseval Relation 704

28*. Hilbert Space 706

29*. Orthogonality with Weight Function 708

30*. Multiple Fourier Series 710

31*. Application to the Equation of Oscillations of a String 711

§ 5. Fourier Transformation 713

32*. Fourier Transform 713

33*. Properties of Fourier Transforms 717

34*. Application to Oscillations of Infinite String 719

CHAPTER XVIII. ELEMENTS OF THE THEORY OF PROBABILITY 721

§ 1. Random Events and Their Probabilities 721

1. Random Events 721

2. Probability 722

3. Basic Properties of Probabilities 725

4. Theorem of Multiplication of Probabilities 727

5. Theorem of Total Probability 729

6*. Formulas for the Probability of HyPotheses 730

7. Disregarding Low-Probability Events 731

§ 2. Random Variables 732

8. Definitions 732

9. Examples of Discrete Random Variables 734

10. Examples of Continuous Random Variables 736

11. Joint Distribution of Several Random Variables 737

12. Functions of Random Variables 739

§ 3. Numerical Characteristics of Random Variables 741

13. The Mean Value 741

14. Properties of the Mean Value 742

15. Variance 744

16*. Correlation 746

17. Characteristic Functions 748

§ 4. Applications of the Normal Law 750

18. The Normal Law as the Limiting One 750

19. Confidence Interval 752

20. Data Processing 754

CHAPTER XIX. COMPUTERS 757

§ 1. Two Classes of Computers 757

1. Analogue Computers 758

2. Digital Computers 762

§ 2. Programming 764

3. Number Systems 764

4. Representing Numbers in a Computer 766

5. Instructions 769

6. Examples of Programming 772

Appendix. Equations of Mathematical Physics 780

1*. Derivation of Some Equations 780

2*. Some Other Equations 783

3*. Initial and Boundary Conditions 784

§ 2. Method of Separation of Variables 786

4*. Basic Example 786

5*. Some Other Problems 791

Bibliography 796

Name Index 798

Subject Index 8OO

List of Symbols 815

this is nice book.i want to pdf link . can you send me

the link is in the post

Missing pages can be found here: https://ufile.io/cw7b7

Thanks, will incorporate this in the file and update

PS: unfortunately, you missed to quote one of the missing pages (i cant recall which, just look around the ones you cited already above..), so I didn’t include it in the archive. If you want, i can try to obtain it (but not sure if still possible, sorry).

ok sure, will check and will also have to update the bookmarks, which will take some time

Hi Tony,

These pages are still missing: 170-171, 220, 284-285, 315, 628-629, 636-637, 758

Hope you find them and update us with these pages. Also if possible scan them at a higher resolution, as current scan of pages given by you somehow are not getting OCRed.

The link: https://ufile.io/cw7b7 it´s expired, only premium users can download it!!!! 😦