Introductory mathematics for engineers: lectures in higher mathematics

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
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
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
§ 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
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 780/750,Black,notBold,notItalic,open,FitPage 2*. Some Other Equations 783
3*. Initial and Boundary Conditions 784/754,Black,notBold,notItalic,open,FitPage § 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