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

Pingback: Mathematical Analysis for Engineers – Krasnov, Kiselev, Makarenko, Shikin — Mir Books | Chet Aero Marine

Thank you for uploading. Please note in Vol II, pages 332-3335 are missing

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Global Director, Engineering Standards

IMI CCI

Tel: +1 949 835 8208

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http://www.imi-critical.com

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We do not have access to a physical copy, have posted the already existing copy.

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Dear Mitr or Dima,

Can you find and upload L.A.Lyusternik’s “The Shortest Lines” (Mir Publishers, 1976).

Thanks for all the efforts!

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It is one of the remaining four from the Little Mathematics Library. So far we could not locate physical/digital copies of these. You can check the rest at

https://mirtitles.org/2012/09/06/little-mathematics-library-taking-stock/

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The link for volume-2 is broken

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It is working fine.

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