Introductory Mathematics for Engineers

Front Cover......Page 1
Title Page......Page 3
Preface 5......Page 5
Contents......Page 8
1. The Subject of Mathematics 19......Page 18
3. Abstractness 20......Page 19
4. Characteristic Features of Higher Mathematics 22......Page 21
5. Mathematics in the Soviet Union 23......Page 22
2. Dimensions of Quantities 25......Page 24
3. Constants and Variables 26......Page 25
4. Number Scale. Slide Rule 27......Page 26
5. Characteristics of Variables 29......Page 28
7. Errors 32......Page 31
8. Writing Approximate Numbers 33......Page 32
9. Addition and Subtraction of Approximate Numbers 34......Page 33
10. Multiplication and Division of Approximate Numbers Remarks 36......Page 35
11. Functional Relation 39......Page 38
12. Notation 40......Page 39
13. Methods of Representing Functions 42......Page 41
14. Graphs of Functions 45......Page 44
15. The Domain of Definition of a Function 47......Page 46
16. Characteristics of Behaviour of Functions 48......Page 47
17. Algebraic Classification of Functions 51......Page 50
18. Elementary Functions 53......Page 52
19. Transforming Graphs 54......Page 53
21. Inverse Functions 58......Page 57
22. Linear Function 60......Page 59
23. Quadratic Function 62......Page 61
24. Power Function 63......Page 62
25. Linear-Fractional Function 66......Page 64
26. Logarithmic Function 68......Page 67
27. Exponential Function 69......Page 68
28. Hyperbolic Functions 70......Page 69
29. Trigonometric Functions 72......Page 71
30. Empirical Formulas 75......Page 74
1. Cartesian Coordinates 78......Page 77
2. Some Simple Problems Concerning Cartesian Coordinates 79......Page 78
3. Polar Coordinates 81......Page 80
4. Equation of a Curve in Cartesian Coordinates 82......Page 81
5. Equation of a Curve in Polar Coordinates 84......Page 83
6. Parametric Representation of Curves and Functions 87......Page 86
7. Algebraic Curves 90......Page 89
8. Singular Cases 92......Page 91
9. Curves of the First Order 94......Page 93
10. Ellipse 96......Page 95
11. Hyperbola 99......Page 98
12. Relationship Between Ellipse, Hyperbola and Parabola 102......Page 101
13. General Equation of a Curve of the Second Order 105......Page 104
1. Infinitesimal Variables 109......Page 108
2. Properties of Infinitesimals 111......Page 110
3. Infinitely Large Variables 112......Page 111
4. Definition 113......Page 112
5. Properties of Limits 115......Page 114
6. Sum of a Numerical Series 117......Page 116
7. Comparison of Infinitesimals 121......Page 120
9. Important Examples 122......Page 121
10. Orders of Smallness 124......Page 123
12. Definition of a Continuous Function 125......Page 124
13. Points of Discontinuity 126......Page 125
14. Properties of Continuous Functions 129......Page 128
15. Some Applications 131......Page 130
1. Some Problems Leading to the Concept of a Derivative 134......Page 133
2. Definition of Derivative 136......Page 135
3. Geometrical Meaning of Derivative 137......Page 136
4. Basic Properties of Derivatives 139......Page 138
5. Derivatives of Basic Elementary Functions 142......Page 141
6. Determining Tangent in Polar Coordinates 146......Page 145
7. Physical Examples 148......Page 147
8. Definition of Differential and Its Connection with Increment 149......Page 148
9. Properties of Differential 152......Page 151
10. Application of Differentials to Approximate Calculations 153......Page 152
11. Derivatives of Higher Orders 155......Page 154
12. Higher-Order Differentials 156......Page 155
13. Indeterminate Forms of the Type $\\dfrac{0}{0}$ 158......Page 157
14. Indeterminate Forms of tl1e Type $\\dfrac{\\infty}{\\infty}$ 160......Page 159
15. Taylor\'s Formula 161......Page 160
16. Taylor\'s Series 163......Page 162
17. Sign of Derivative 165......Page 164
18. Points of Extremum 166......Page 165
19. The Greatest and the Least Values of a Function 168......Page 167
20. Intervals of Convexity of a Graph and Points of Inflection 173......Page 172
21. Asymptotes of a Graph 174......Page 173
22. General Scheme for Investigating a Function and Constructing Its Graph 175......Page 174
1. Introduction 179......Page 178
2. Cut-and-Try Method. Method of Chords. Method of Tangents 181......Page 180
3. Iterative Method 185......Page 184
4. Formula of Finite Increments 187......Page 186
5*. Small Parameter Method 189......Page 188
6. Lagrange\'s Interpolation Formula 191......Page 190
7. Finite Differences and Their Connection with Derivatives 192......Page 191
8. Newton\'s Interpolation Formulas 196......Page 195
9. Numerical Differentiation 198......Page 187
1. Definition 200......Page 199
2. Properties 201......Page 200
3. Expanding a Determinant in Minors of Its Row or Column 203......Page 202
4. Basic Case 206......Page 205
5. Numerical Solution 208......Page 207
6. Singular Case 209......Page 208
1. Scalar and Vector Quantities 212......Page 211
2. Addition of Vectors 213......Page 212
4. Multiplying a Vector by a Scalar 215......Page 214
5. Linear Combination of Vectors 216......Page 215
6. Projection of Vector on Axis 219......Page 218
7. Scalar Product 220......Page 219
8. Properties of Scalar Product 221......Page 220
9. Cartesian Coordinates in Space 222......Page 221
10. Some Simple Problems Concerning Cartesian Coordinates 223......Page 222
11. Orientation of Surface and Vector of an Area 227......Page 226
12. Vector Product 228......Page 227
13. Properties of Vector Products 230......Page 229
14*. Pseudovectors 233......Page 232
15. Triple Scalar Product 235......Page 234
16. Triple Vector Product 236......Page 235
17. Concept of Linear Space 237......Page 236
18. Examples 239......Page 238
3. Number Systems 764......Page 0
20. Concept of Euclidean Space 244......Page 243
21. Orthogonality 245......Page 244
23. Vector Functions of Scalar Argument 248......Page 247
24. Some Notions Related to the Second Derivative 251......Page 250
25. Osculating Circle 252......Page 251
26. Evolute and Evolvent 255......Page 254
1. Complex Plane 259......Page 258
2. Algebraic Operations on Complex Numbers 261......Page 260
3. Conjugate Complex Numbers 263......Page 262
4. Euler\'s Formula 264......Page 263
5. Logarithms of Complex Numbers 266......Page 265
6. Definition and Properties 267......Page 266
7*. Applications to Describing Oscillations 269......Page 268
8. Factorization of a Polynomial 271......Page 270
9*. Numerical Methods of Solving Algebraic Equations 273......Page 272
10. Decomposition of a Rational Fraction into Partial Rational Fractions 277......Page 276
11*. Some General Remarks on Functions of a Complex Variable 280......Page 279
1. Methods of Representing 283......Page 282
2. Domain of Definition 286......Page 285
3. Linear Function 287......Page 286
4. Continuity and Discontinuity 288......Page 287
6. Methods of Representing 291......Page 290
8. General Case 292......Page 291
9. Concept of Field 293......Page 292
10. Basic Definitions 294......Page 293
11. Total Differential 296......Page 295
12. Derivative of Composite Function 298......Page 297
13. Derivative of Implicit Function 300......Page 299
14. Definitions 303......Page 302
15. Equality of Mixed Derivatives 304......Page 303
16. Total Differentials of Higher Order 305......Page 304
1. Coordinate Systems in Space 307......Page 306
2*. Degrees of Freedom 309......Page 308
4. Cylinders, Cones and Surfaces of Revolution 314......Page 313
5. Curves In Space 316......Page 315
6. Parametric Representation of Surfaces in Space. Parametric Representation of Functions of Several Variables 317......Page 316
7. Algebraic Surfaces of the First Order 319......Page 318
8. Ellipsoids 322......Page 321
9. Hyperboloids 324......Page 323
10. Paraboloids 326......Page 325
11. General Review of the Algebraic surfaces of the second order 327......Page 326
1. Definitions 329......Page 328
2. Operations on Matrices 331......Page 330
3. Inverse Matrix 333......Page 332
4. Eigenvectors and Eigenvalues of a Matrix 335......Page 334
7. Transformation of the Matrix of a Linear Mapping When the Basis Is Changed 347......Page 346
8. The Matrix of a Mapping Relative to the Basis Consisting of Its Eigenvectors 350......Page 349
9. Transforming Cartesian Basis 352......Page 351
10. Symmetric Matrices 353......Page 352
11. Quadratic Forms 355......Page 354
12. Simplification of Equations of Second-Order Curves and Surfaces 357......Page 356
13*. General Notions 358......Page 357
14*. Non Linear Mapping in the Small 360......Page 359
15*. Functional Relation Between Functions 362......Page 361
1. Directional Derivative. Gradient 365......Page 364
2. Level Surfaces 368......Page 367
3. Implicit Functions of Two Independent Variables 370......Page 369
4. Plane Fields 371......Page 370
5. Envelope of One-Parameter Family of Curves 372......Page 371
6. Taylor\'s Formula for a Function of Several Variables 374......Page 373
7. Extremum 375......Page 374
8. The Method of Least Squares 380......Page 379
9*. Curvature of Surfaces 381......Page 380
10. Conditional Extremum 384......Page 383
11. Extremum with Unilateral Constraints 388......Page 387
12*. Numerical Solution of Systems of Equations 390......Page 389
1. Basic Definitions 393......Page 392
2. The Simplest Integrals 394......Page 393
3. The Simplest Properties of an Indefinite Integral 397......Page 396
4. Integration by Parts 399......Page 398
5. Integration by Change of Variable (by Substitution) 402......Page 401
§ 2. Standard Methods of Integration 404......Page 403
6. Integration of Rational Functions 405......Page 404
7. Integration of Irrational Functions Involving Linear and Linear-Fractional Expressions 407......Page 406
8. Integration of Irrational Expressions Containing Quadratic Trinomials 408......Page 407
9. Integrals of Binomial Differentials 411......Page 410
lO. Integration of Functions Rationally Involving Trigonometric Functions 412......Page 411
11. General Remarks 415......Page 414
1. Examples Lending to the Concept of Definite Integral 417......Page 416
3. Relationship Between Definite Integral and Indefinite Integral 426......Page 422
4. Basic Properties of Definite Integral 433......Page 432
6. Two Schemes of Application 436......Page 435
7. Differential Equations with Variables Separable 437......Page 436
8. Computing Areas of Plane Geometric Figures 443......Page 438
9. The Arc Length of a Curve 445......Page 444
10. Computing Volumes of Solids 447......Page 446
12. General Remarks 448......Page 447
13. Formulas of Numerical Integration 450......Page 449
§ 4. Improper Integrals 454......Page 453
14. Integrals with Infinite Limits of Integration 455......Page 454
15. Basic Properties of Integrals with Infinite Limits 464......Page 465
16. Other Types of Improper Integral 468......Page 469
17*. Gamma Function 468......Page 467
18*. Beta Function 471......Page 470
19*. Principal Value of Divergent Integral 473......Page 472
20*. Proper Integrals 474......Page 473
21*· Improper Integrals 476......Page 475
§ 6. Line Integrals of Integration 478......Page 477
22. Line Integrals of the First Type 482......Page 481
23. Line Integrals of the Second Type 484......Page 483
25*. Delta Function 488......Page 487
26*. Application to Constructing Influence Function 492......Page 491
27*. Other Generalized Functions 495......Page 494
1. Examples 497......Page 496
2. Basic Definitions 498......Page 497
3. Geometric Meaning 500......Page 499
4. Integrable Types of Equations 503......Page 502
5*. Equation for Exponential Function 506......Page 505
6. Integrating Exact Differential Equations 509......Page 508
7. Singular Points and Singular Solutions 512......Page 511
8. Equations Not Solved for the Derivative 516......Page 515
9. Method of Integration by Means of Differentiation 517......Page 516
13*. First Integrals 526......Page 518
11*. Connection Between Higher-Order Equations and Systems of First-Order Equations 521......Page 520
12*. Geometric Interpretation of System of First-Order Equations 522......Page 521
16*. Boundary-Value Problems 535......Page 527
15. Non-Homogeneous Equations 530......Page 529
17. Homogeneous Equations 541......Page 540
18. Non-Homogeneous Equations with Right-Hand Sides of Special Form 545......Page 544
19. Euler\'s Equations 548......Page 547
20*. Operators and the Operator Method of Solving Differential Equations 549......Page 545
21. Systems of Linear Equations 553......Page 552
22*. Applications to Testing Lyapunov Stability of Equilibrium State 558......Page 557
23. Iterative Method 562......Page 561
24*. Application of Taylor\'s Series 564......Page 563
25. Application of Power Series with Undetermined coefficients 565......Page 564
26*. Bessel\'s Functions 566......Page 565
27*. Small Parameter Method 569......Page 568
28*. General Remarks on Dependence of Solutions on Parameters 572......Page 571
29*. Methods of Minimizing Discrepancy 575......Page 574
30*. Simplification Method 576......Page 575
31. Euler\'s Method  578......Page 577
32. Runge-Kutta Method 580......Page 579
33. Adams Method  582......Page 581
34. Milne\'s Method  583......Page 582
1. Some Examples Leading to the Notion of a Multiple Integral 585......Page 584
2. Definition of a Multiple Integral 586......Page 585
3. Basic Properties of Multiple Integrals 587......Page 586
4. Methods of Applying Multiple Integrals 589......Page 588
5. Geometric Meaning of an Integral Over a Plane Region 591......Page 590
6*. Basic Example. Mass and Its Density 592......Page 591
7*. Quantities Distributed in Space 594......Page 593
8. Integral Over Rectangle 596......Page 595
9. Integral Over an Arbitrary Plane Region 599......Page 598
10. Integral Over an Arbitrary Surface 602......Page 601
11. Integral Over a Three-Dimensional Region 604......Page 603
12. Passing to Polar Coordinates in Plane 605......Page 604
13. Passing to Cylindrical and Spherical Coordinates 606......Page 605
14*. Curvilinear Coordinates in Plane 608......Page 607
15*. Curvilinear Coordinates in Space 611......Page 610
16*. Coordinates on a Surface 612......Page 611
17*. Improper Integrals 615......Page 614
18*. Integrals Dependent on a Parameter 617......Page 616
19*. Integrals with Respect to Measure. Generalized Functions 620......Page 619
20*. Multiple Integrals of Higher Order 622......Page 621
21*. Vector Lines 626......Page 625
22*. The Flux. of a Vector Through a Surface 627......Page 626
23*. Divergence 629......Page 630
24*. Expressing Divergence in Cartesian Coordinates 632......Page 631
26*. Rotation 634......Page 633
27. Green\'s Formula. Stokes\' Formula 638......Page 637
28*. Expressing Differential Operations on Vector Fields in a Curvilinear Orthogonal Coordinate System 641......Page 640
29*. General Formula for Transforming Integrals 642......Page 641
1. Positive Series 645......Page 644
2. Series with Terms of Arbitrary Signs 650......Page 649
3. Operations on Series 652......Page 651
4*. Speed of Convergence of a Series 654......Page 653
5. Series with Complex, Vector and Matrix Terms 658......Page 657
6. Multiple Series 659......Page 658
7. Deviation of Functions 661......Page 660
8. Convergence of a Functional Series 662......Page 661
9. Properties of Functional Series 664......Page 663
10. Interval of Convergence 666......Page 665
11. Properties of Power Series 667......Page 666
12. Algebraic Operations on Power Series 671......Page 670
13. Power Series as a Taylor Series 675......Page 674
14. Power Series with Complex Terms 676......Page 675
15*. Bernoullian Numbers 677......Page 676
16*. Applying Series to Solving Difference Equations 678......Page 677
17*. Multiple Power Series 680......Page 679
18*. Functions of Matrices 681......Page 680
19*. Asymptotic Expansions 685......Page 683
20. Orthogonality 686......Page 685
21. Series in Orthogonal Functions 689......Page 688
22. Fourier Series 690......Page 689
23. Expanding a Periodic Function 695......Page 694
24*. Example. Bessel\'s Functions as Fourier Coefficients 697......Page 696
25. Speed of Convergence of a Fourier Series 698......Page 697
26. Fourier Series in Complex Form 702......Page 701
27*. Parseval Relation 704......Page 703
28*. Hilbert Space 706......Page 705
29*. Orthogonality with Weight Function 708......Page 707
30*. Multiple Fourier Series 710......Page 709
31*. Application to the Equation of Oscillations of a String 711......Page 710
32*. Fourier Transform 713......Page 712
33*. Properties of Fourier Transforms 717......Page 716
34*. Application to Oscillations of Infinite String 719......Page 718
1. Random Events 721......Page 720
2. Probability 722......Page 721
3. Basic Properties of Probabilities 725......Page 723
4. Theorem of Multiplication of Probabilities 727......Page 725
5. Theorem of Total Probability 729......Page 728
6*. Formulas for the Probability of HyPotheses 730......Page 729
7. Disregarding Low-Probability Events 731......Page 730
8. Definitions 732......Page 731
9. Examples of Discrete Random Variables 734......Page 733
10. Examples of Continuous Random Variables 736......Page 735
11. Joint Distribution of Several Random Variables 737......Page 736
12. Functions of Random Variables 739......Page 738
13. The Mean Value 741......Page 740
14. Properties of the Mean Value 742......Page 741
15. Variance 744......Page 743
16*. Correlation 746......Page 745
17. Characteristic Functions 748......Page 747
18. The Normal Law as the Limiting One 750......Page 749
19. Confidence Interval 752......Page 751
20. Data Processing 754......Page 753
§ 1. Two Classes of Computers 757......Page 756
1. Analogue Computers 758......Page 757
2. Digital Computers 762......Page 759
4. Representing Numbers in a Computer 766......Page 763
5. Instructions 769......Page 766
6. Examples of Programming 772......Page 769
Appendix. Equations of Mathematical Physics 780......Page 779
1*. Derivation of Some Equations 780/750,Black,notBold,notItalic,open,FitPage         2*. Some Other Equations 783......Page 782
4*. Basic Example 786......Page 785
5*. Some Other Problems 791......Page 790
Bibliography 796......Page 795
Name Index 798......Page 797
Subject Index 8OO......Page 799
List of Symbols 815......Page 814