In this post, we will see the two volume set of books titled Mathematical Analysis For Engineers Volumes 1 & 2 by M. Krasnov, A. Kiselev, G. Makarenko, E. Shikin .
About the book
This two-volume book was written for students of technical colleges who have had the usual mathematical training. It contains just enough information to continue with a wide variety of engineering disciplines. It covers analytic geometry and linear algebra, differential and integral calculus for functions of one and more variables, vector analysis, numerical and functional series (including Fourier series), ordinary differential equations, functions of a complex variable, Laplace and Fourier transforms, and equations of mathematical physics. This list itself demonstrates that the book covers the material for both a basic course in higher mathematics and several specialist sections that are important for applied problems. Hence, it may be used by a wide range of readers. Besides students in technical colleges and those starting a mathematics course, it may be found useful by engineers and scientists who wish to refresh their knowledge of some aspects of mathematics.
We tried to give the fundamental material concisely and without distracting detail. We concentrated on the presentation of the basic ideas of linear algebra and analysis to make it detailed and as comprehensible as possible. Mastery of these ideas is a requirement to understand the later material.
The many examples also serve this aim. The examples were written to help students with the mechanics of solving typical problems. More than 600 diagrams are simple illustrations, clear enough to demonstrate the ideas and statements convincingly, and can be fairly easily reproduced.
In addition to the examples, we have included several carefully selected problems and exercises (around 1000) which should be of interest to those pursuing an independent mathematics course. The problems have the form of moderately sized theorems. They are very simple but are good training for those learning the fundamental ideas.
The book was translated from Russian by Alexander Yastrebov and was published by Mir in 1990.
Credits to the original uploader.
You can get Volume 1 here and Volume 2 here.
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Contents Volume 1
Preface 9
Chapter 1 An Introduction to Analytic Geometry 11
1.1 Cartesian Coordinates 11
1.2 Elementary Problems of Analytic Geometry 14
1.3 Polar Coordinates 18
1.4 Second- and Third-Order Determinants 19
Chapter 2 Elements of Vector Algebra 24
2.1 Fixed Vectors and Free Vectors 24
2.2 Linear Operations on Vectors 26
2.3 Coordinates and Components of a Vector 30
2.4 Projection of a Vector onto an Axis 33
2.5 Scalar Product of Two Vectors 34
2.6 Vector Product of Two Vectors 39
2.7 Mixed Products of Three Vectors 43
Exercises 45
Answers 46
Chapter 3 The Line and the Plane 47
3.1 The Plane 47
3.2 Straight Line in a Plane 51
3.3 Straight Line in Three-Dimensional Space 55
Exercises 60
Answers 62
Chapter 4 Curves and Surfaces of the Second Order 63
4.1 Changing the Axes of Coordinates in a Plane 63
4.2 Curves of the Second Order 66
4.3 The Ellipse 67
4.4 The Hyperbola 71
4.5 The Parabola 77
4.6 Optical Properties of Curves of the Second Order 79
4.7 Classification of Curves of the Second Order 83
4.8 Surfaces of the Second Order 89
4.9 Classification of Surfaces 90
4.10 Standard Equations of Surfaces of the Second Order 95
Exercises 102
Answers 102
Chapter 5 Matrices. Determinants. Systems of Linear Equations 103
5.1 Matrices 103
5.2 Determinants 122
5.3 Inverse Matrices 133
5.4 Rank of a Matrix 139
5.5 Systems of Linear Equations 143
Exercises 165
Answers 167
Chapter 6 Linear Spaces and Linear Operators 168
6.1 The Concept of Linear Space 168
6.2 Linear Subspaces 170
6.3 Linearly Dependent Vectors 174
6.4 Basis and Dimension 175
6.5 Changing a Basis 181
6.6 Euclidean Spaces 183
6.7 Orthogonalization 185
6.8 Orthocompliments of Linear Subspaces 189
6.9 Unitary Spaces 191
6.10 Linear Mappings 192
6.11 Linear Operators 197
6.12 Matrices of Linear Operators 200
6.13 Eigenvalues and Eigenvectors 205
6.14 Adjoint Operators 209
6.15 Symmetric Operators 211
6.16 Quadratic Forms 213
6.17 Classification of Curves and Surfaces of the Second Order 221
Exercises 227
Answers 228
Chapter 7 An Introduction to Analysis 229
7.1 Basic Concepts 229
7.2 Sequences of Numbers 239
7.3 Functions of One Variable and Limits 247
7.4 Infinitesimals and Infinities 258
7.5 Operations on Limits 266
7.6 Continuous Functions. Continuity at a Point 272
7.7 Continuity on a Closed Interval 283
7.8 Comparison of Infinitesimals 288
7.9 Complex Numbers 294
Exercises 302
Answers 304
Chapter 8 Differential Calculus. Functions of One Variable 305
8.1 Derivatives and Differentials 305
8.2 Differentiation Rules 316
8.3 Differentiation of Composite and Inverse Functions 324
8.4 Derivatives and Differentials of Higher Orders 332
8.5 Mean Value Theorems 339
8.6 L’Hospital’s Rule 344
8.7 Tests for Increase and Decrease of a Function on a Closed Interval and at a Point 349
8.8 Extrema of a Function. Maximum and Minimum of a Function on a Closed Interval 352
8.9 Investigating the Shape of a Curve. Points of Inflection 362
8.10 Asymptotes of a Curve 367
8.11 Curve Sketching 373
8.12 Approximate Solution of Equations 381
8.13 8.Taylor’s Theorem 385
8.14 Vector Function of a Scalar Argument 396
Exercises 401
Answers 403
Chapter 9 Integral Calculus. The Indefinite Integral 409
9.1 Basic Concepts and Definitions 409
9.2 Methods of Integration 414
9.3 Integrating Rational Function 424
9.4 Integrals Involving Irrational Functions 435
9.5 Integrals Involving Trigonometric Functions 445
Exercises 450
Answers 453
Chapter 10 Integral Calculus. The Definite Integral 456
10.1 Basic Concepts and Definitions 456
10.2 Properties of the Definite Integral 461
10.3 Fundamental Theorems for Definite Integrals. 467
10.4 Evaluating Definite Integrals 472
10.5 Computing Areas and Volumes by Integration 476
10.6 Computing Arc Lengths by Integration 488
10.7 Applications of the Definite Integral 495
10.8 Numerical Integration 498
Exercises 503
Answers 505
Chapter 11 Improper Integrals 506
11.1 Integrals with Infinite Limits of Integration 506
11.2 Integrals of Nonnegative Functions 511
11.3 Absolutely Convergent Improper Integrals 514
11.4 Cauchy Principal Value of the Improper Integrals 519
11.5 Improper Integrals of Unbounded Functions 520
11.6 Improper Integrals of Unbounded Nonnegative Functions. Convergence Tests 523
11.7 Cauchy Principal Value of the Improper Integral Involving Unbounded Functions 525
Exercises 526
Answers 527
Chapter 12 Functions of Several Variables 529
12.1 Basic Notions and Notation 529
12.2 Limits and Continuity 533
12.3 Partial Derivatives and Differentials 538
12.4 Derivatives of Composite Functions 545
12.5 Implicit Functions 550
12.6 Tangent Planes and Normal Lines to a Surface 555
12.7 Derivatives and Differentials of Higher Orders 558
12.8 Taylor’s Theorem 562
12.9 Extrema of a Function of Several Variables 566
Exercises 580
Answers 583
Appendix I Elementary Functions 587
Index 596
Contents Volume 2
Preface 11
Chapter 13 Number Series 13
13.1 Definition. Sum of a Series 13
13.2 Operations on Series 15
13.3 Tests for Convergence of Series 18
13.4 Alternating Series. Leibniz Test 30
13.5 Series of Positive and Negative Terms. Absolute and Conditional Convergence 32
Exercises 35
Answers 37
Chapter 14 Functional Series 38
14.1 Convergence Domain and Convergence Interval 38
14.2 Uniform Convergence 40
14.3 Weierstrass Test 43
14.4 Properties of Uniformly Convergent Functional Series 45
Exercises 50
Answers 50
Chapter 15 Power Series 51
15.1 Abel’s Theorem. Interval and Radius of Convergence for Power Series 51
15.2 Properties of Power Series 56
15.3 Taylor’s Series 59
Exercises 70
Answers 71
Chapter 16 Fourier Series 73
16.1 Trigonometric Series 73
16.2 Fourier Series for a Function with Period 2𝜋 76
16.3 Sufficient Conditions for the Fourier Expansion of a Function 78
16.4 Fourier Expansions of.Odd and Even Functions 82
16.5 Expansion of a Function Defined on the Given Interval into series of Sines and Cosines 86
16.6 Fourier Series for a Function with Arbitrary Period 88
16.7 Complex Representation of Fourier Series 93
16.8 Fourier Series in General Orthogonal Systems of Functions 96
Exercises 104
Answers 105
Chapter 17 First-Order Ordinary Differential Equations 106
17.1 Basic Notions. Examples 106
17.2 Solution of the Cauchy Problem for First-Order Differential Equations 109
17.3 Approximate Methods of Integration of the Equation y’= f(x y) 113
17.4Some Equations Integrable by Quadratures 118
17.5 Riccati Equation 135
17.6 Differential Equations Insolvable for the Derivative 136
17.7 Geometrical Aspects of First-Order Differential Equations. Orthogonal Trajectories 142
Exercises 144
Answers 145
Chapter 18 Higher-Order Differential Equations 147
18.1 Cauchy Problem 147
18.2 Reducing the Order of Higher-Order Equations 149
18.3 Linear Homogeneous Differential Equations of Order n 153
18.4 Linearly Dependent and Linearly Independent Systems of Functions 155
18.5 Structure of General Solution of Linear Homogeneous Differential Equation 160
18.6 Linear Homogeneous Differential Equations with Constant Coefficients 164.
18.7 Equations Reducible to Equations with Constant Coefficients 172
18.8 Linear Inhomogeneous Differential Equations 173
18.9 Integration of Linear Inhomogeneous Equation by Variation of Constants 176
18.10 Inhomogeneous Linear Differential Equations with Constant Coefficients 180
18.11 Integration of Differential Equations Using Rower Series and Generalized Power Series 188
18.12 Bessel Equation. Bessel Functions 190
Exercises 201
Answers 201
Chapter 19 Systems of Differential Equations 203
19.1 Essentials. Definitions 203
19.2 Methods of Integration of Systems of Differential Equations 206
19.3 Systems of Linear Differential Equations 211
19.4 Systems of Linear Differential Equations With Constant Coefficients aj
Exercises 224
Answers 224
Chapter 20 Stability Theory 225
20.1 Preliminaries 225
20.2 Stability in the Sense} of Lyapunov. Basic Concepts and Definitions 227
20.3 Stability of Autonomous Systems. Simplest Types of Stationary Points 233
20.4 Method of Lyapunov’s Functions 244
20.5 Stability in First (Linear) Approximation, 248
Exercises 253
Answers 254
Chapter 21 Special Topics of Differential Equations 255
21.1 Asymptotic Behaviour of Solutions of Differential Equations as x → ∞ 255
21.2 Perturbation Method 257
21.3 Oscillations of Solutions .of Differential Equations 261
Exercises 264
Answers 264
Chapter 22 Multiple Integrals. Double Integral 265
22.1 Problem Leading to the Concept of Double Integral 265
22.2 Main Properties of Double Integral 268
22.3 Double Integral Reduced to Iterated Integral 270
22.4 Change of Variables in Double Integral 278
22.5 Surface Area. Surface Integral 286
22.6 Triple Integrals 292
22.7 Taking Triple Integral in Rectangular Coordinates 294
22.8 Taking Triple Integral in Cylindrical and Spherical Coordinates 296
22.9 Applications of Double and Triple Integrals 302
22.10 Improper Multiple Integrals over Unbounded Domains 307
Exercises 309
Answers 312
Chapter 23 Line Integrals 313
23.1 Line Integrals of the First Kind 313
23.2 Line Integrals of the Second Kind 318
23.3 Green’s Formula 322
23.4 Applications of Line Integrals 327
Exercises 331
Answers 333
Chapter 24 Vector Analysis 334
24.1 Scalar Field. Level Surfaces and Curves. Directional Derivative 334
24.2 Gradient of a Scalar Field 339
24.3 Vector Field. Vector Lines and Their Differential Equations 344
24.4 Vector Flux Through a Surface and Its Properties 349
24.5 Flux of a Vector Through an Open Surface 354
24.6 Flux of a Vector Through a Closed Surface. Ostrogradsky-Gauss Formula 363
24.7 Divergence of a Vector Field 371
24.8 Circulation of a Vector Field. Curl of a Vector. Stokes Theorem 378
24.9 Independence of the Line Integral of Integration Path 386
24.10 Potential Field 391
24.11 Hamiltonian 398
24.12 Differential Operations of the Second Order. Laplace Operator 402
24.13 Curvilinear Coordinates 406
24.14 Basic Vector Operations in Curvilinear Coordinates 408
Exercises 416
Answers 419
Chapter 25 Integrals Depending on Parameter 420
25.1 Proper Integrals Depending on Parameter 420
25.2. Improper Integrals Depending on Parameter 425
25.3 Euler Integrals. Gamma Function. Beta Function 431
Exercises 436
Answers 438
Chapter 26 Functions of a Complex Variable 441 .
26.1 Essentials. Derivative. Cauchy-Riemann Equations 441
26.2 Elementary Functions of a Complex Variable 453
26.3 Integration with Respect to a Complex: Argument. Cauchy Theorem. Cauchy Integral. Formula 461
26.4 Complex Power Series. Taylor Series 476
26.5 Laurent Series. Isolated Singularities and Their Classification 491
26.6 Residues. Basic Theorem on Residues. Application of Residues to Integrals 503
Exercises 519
Answers 522
Chapter 27 Integral Transforms. Fourier Transforms 524
27.1 Fourier Integral 524 i
27.2 Fourier Transform. Fourier Sine and Cosine Transforms 528
27.3 Properties of the Fourier Transform 535
27.4 Applications 539
27.5 Multiple Fourier Transforms 543
Exercises 544
Answers 545
Chapter 28 Laplace Transform 546
28.1 Basic Definitions 546
28.2 Properties of Laplace Transform 551
28.3 Inverse Transform 560
28.4 Applications of Laplace Transform (Operational Calculus) 565
Exercises 572
Answers 573
Chapter 29 Partial Differential Equations 575
29.1 Essentials. Examples 575
29.2 Linear Partial Differential Equations. Properties of Their Solutions 577
29.3 Classification of Second- Order Linear Differential Equations in Two Independent Variables 579
Exercises 583
Answers 584
Chapter 30 Hyperbolic Equations 585
30.1 Essentials 585
30.2 Solution of the Cauchy Problem (Initial Value Problem) for an Infinite String 587
30.3 Examination of the D’Alembert Formula 591
30.4 Well-Posedness of a Problem. Hadamard’s Example of Ill-Posed Problem 594
30.5 Free Vibrations of a String Fixed at Both Ends. Fourier Method 598.
30.6 Forced Vibrations of a String Fixed at Both Ends 606
30.7 Forced Vibrations of a String with Unfixed Ends 611
30.8 General Scheme of the Fourier Method 613
30.9 Uniqueness of Solution of a Mixed Problem 621
30.10 Vibrations of a Round Membrane 623
30.11 Application of Laplace Transforms to Solution of Mixed Problems 627
Exercises 631
Answers 632
Chapter 31 Parabolic Equations 633
31.1 Heat Equation 633
31.2 Cauchy’ Problem for Heat Equation 634
31.3 Heat Propagation in a Finite Rod 640
31.4 Fourier Method For Heat Equation 643
Exercises 649
Answers 649
Chapter 32 Elliptic Equations 650
32.1 Definitions. Formulation of Boundary Problems 650
32.2 Fundamental Solution of Laplace Equation 652
32.3 Green’s Formulas 653
32.4 Basic Integral Green’s Formula 654
32.5 Properties of Harmonic Functions 657 :.
32.6 Solution of the Dirichlet Problem for a Circle Using the Fourier Method 661
32.7 Poisson: Integral 664
Exercises 666
Answers 666
Appendix II Conformal Mappings 667
Index 693
