Updates will also be available via this channel in the future. We will use hashtag #mirtitles so keep a lookout for it.

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Happy reading!

]]>in Science and Technology in the USSR

**About the book (From the Preface)**

The modern theory of nonlinear phenomena is going through a period of explosive growth. Each “invention” in the field is disseminated rapidly, generating general interest and unexpected applications. This was for instance the case with solitons, strange attractors, and stochasticity. That scientists from each of the different branches of the theory of nonlinear phenomena should cooperate needs no proof, and the value of this cooperation could be seen at a conference held in Kiev on nonlinear and turbulent processes in physics.* The same approach has been adopted for this collection, which includes articles on regular nonlinear phenomena (vortices, solitons, auto-waves) and some on stochasticity and turbulence. In addition, the collection includes two mathematical articles that develop the Kolmogorov-Amold-Moser theory, the importance of which for the theory of nonlinear dynamic systems is undoubted.

The articles are to some degree grouped around the nonlinear problems of plasma physics and hydrodynamics, in their broadest sense.

Hydrodynamics and plasma physics are traditional sources of exciting problems and ideas for the physics of nonlinear phenomena. We need only recall the classical examples of the discovery of solitons in shallow water, the exact integrability of the Korteweg-de Vries equation, and the discoveries of a strange attractor in the Lorcntz system and stochasticity in Hamiltonian systems with small numbers of degrees of freedom.

These ideas and results, all stimulated by problems in hydrodynamics and plasma physics, quickly gained more general significance for physics as a whole. In turn, many new and efficient techniques are tested on such traditionally difficult subjects as turbulence. For instance, there is the recent application to turbulence of the renormalization group methods, which were successfully employed first in field theory and the physics of phase transitions.

Finally, all this culminates in general fund of knowledge in the

physics of nonlinear phenomena. I hope that the publication of

this collection will advance the progress of physics.

The book was translated from the Russian by Valerii Ilyushchenko, and was first published by Mir in 1986.

Many thanks to *Akbar Azimi* for providing the raw 2-in-1 scans. We cleaned and OCRed the scans.

This book has an essay by A. Zhabotinsky of the Belousov–Zhabotinsky reaction

**Contents**

Preface 7

**Vortices In Plasma and Hydrodynamics by A.B. Mikhailovskii**

Introduction 8

Nonlinear Equations for Rossby Waves 12

The Simplest Nonlinear Equations for Drift Waves 14

Vector Vortical Structures 16

Scalar Vortical Structures 18

Further Development of Concepts Concerning Electrostatic Vortices in Plasma 19

Electromagnetic Vortices 22

Conclusion 24

References 28

**Oscillations and Bifurcations in Reversible Systems by V. I. Arnol’d and M. B. Sevryuk 31 **

Introduction 31

Reversible Mappings 32

Reversible Flows 33

Integrable Reversible Mappings and Vector Fields 34

Kolmogorov’s Tori 35

Weak Reversibility 37

The Local Theory 37

Weak Reversibility in a Local Situation 42

Periodic Solutions 42

Kolmogorov’s Tori for Additional “Even” Coordinates 44

The Local Theory for Additional “Even” Coordinates 45

Application to Reversible Equations 48

Kon-Autonomour Reversible Systems 49

The Lyapunov-Devaney Theorem 51

The Resonance 1:1 52

Further Resonances l:N (N > 2) 54

References 83

**Regular and Chaotic Dynamics of Particles In a Magnetic Field by R. Z. Sagdeev and C. U. Zaslavskii 85 **

Introduction 65

Equations of Motion 67

The Resonances of Longitudinal Motion 69

The Overlapping of the Resonances of Longitudinal Motion 72

A Kinetic Description 75

Equations of Transverse Motion 78

The Resonances of High-Energy Particles 83

Resonances in a Weak Magnetic Field 84

Generalization to a Wave Packet 85

A Kinetic Description of Transverse Motion 86

Quasi-Resonance Particles 88

References 92

**The Renormalization Group Method and Kolmogorov-Arnold-Moser Theory by K. M. Khanin and Ya. G. Sinai 93**

Introduction 93

Rectification of the Nonlinear Rotation of a Circle 97

Construction of Invariant KAM Curves by the Renormalization

Group Method 110

References 118

**Nonlinear Problems of Turbulent Dynamo by Ya B, Zel’dovich and A. A. Rukmalkin 119**

Introduction 119

Nonlinear Mean Field Dynamo 121

MHD Turbulence 130

References 135

Problems of the Theory of Strong Turbulence and Topological Solitons by R. Z. Sagdeev, S. S. Molseev, A. V. Tur, and V. V. Yanovskii 137

Introduction 137

The Scaling Group and Functional Method 140

“Null-Modes” and the Self-Similar Spectrum 152

Invariant Properties of Hydrodynamic Models and Topological Solitons 163

References 180

Self-Oscillations and Auto-Waves in Chemical Systems by A. M. Zhabotinskii 183

Introduction 183

Experimental Studies 184

Theoretical Studies 195

Conclusion 207

References 208

Auto-Waves In Biologically Active Media by V. I. Krinskii 210

Introduction 210

Mathematical Description 211

Local Sources of Auto-Waves 213

Cardiac Disorders 215

Mathematical Simulation of Auto-Wave Sources 216

A Chemically Active Medium 216

New Auto-Wave Modes 217

Wave Sources in Three-Dimensional Active Media 217

The Effect of Medium Parameters on Auto-Wave Sources 218

An Anomalous Reverberator 219

Theoretical Studies of Reverberators 220

About the book

We may distinguish at least the following live sufficiently independent categories of vibratory processes differing in their nature:

free vibrations, i.e., vibrations which are performed by a mechanical system having no energy supply from outside if the system is disturbed from its position of equilibrium and then released;

critical states of rotating shafts and rotors which consist in a sudden increase in the deflections of their axes at definite speeds of rotation (or in definite ranges of speeds);

forced vibrations which result when the mechanical system is acted on by fluctuating external forces (driving forces);

parametric vibrations caused by periodic variations of the para meters of a system (for example, its stiffness);

self-excited vibrations, i.e., vibratory processes which are maintained by constant sources of energy of a non-vibratory nature.

Each of these categories of vibratory processes is discussed in the appropriate chapter.

The book was translated from the Russian by M. Konyaeva and was published by Mir in 1971.

Many thanks to *Akbar Azimi* for providing the raw scan 2-in-1 page scans. We cleaned the book with OCR.

**Contents**

Notation. 7

Introduction. 9

**CHAPTER I. FUNDAMENTALS. 11 **

1. Number of Degrees of Freedom of an Elastic System. 11

2. Classification of Forces. 15

3. Methods for Setting Up Equations of Motion in the General Case. 22

**CHAPTER II. Free vibrations . 24 **

4. Linear Systems of One Degree of Freedom Without Inelastic Resistances. 24

5. Effect of Inelastic Resisting Forces on Free Vibrations

of Linear Systems of One Degree of Freedom. 56

6. Undamped Systems of One Degree of Freedom with Non-Linear Restoring Forces. 68

7. Linear Systems of Several Degrees of Freedom. 87

8. Vibrations of Bars of Uniform Section (Exact Solution) 118

9. Vibrations of Ban of Variable Section. 138

10. Two-Dimensional Vibrations of Disks. 147

11. Flexural Vibrations of Disks. 153

12. Flexural Vibrations of Rectangular Plates. 157

**CHAPTER III. CRITICAL STATES OF ROTATING SHAFTS AND ROTORS 161 **

13. Shaft with One Disk. 161

14. Effect of Friction. 174

15. Automatic Balancing of Rotating Shafts. 185

16. Critical States of Helicopter Rotors. 187

17. Shalt with Several Disks. Rigid Rotor on Elastic Supports 190

**CHAPTER IV. FORCED VIBRATIONS. 193 **

18. Linear Systems of One Degree of Freedom without Inelastic Resistances. 193

19. Linear Systems of One Degree of Freedom with Inelastic Resisting Forces. 226

20. Systems with Non-Linear Restoring Forces (Single Degree of Freedom). 241

21. Linear Systems of Several Degrees of Freedom. 249

22. Linear Systems with Distributed Parameters. 269

**CHAPTER V. PARAMETRIC VIBRATIONS. 279 **

23. Basic Equation. 279

24. Cases of Periodic Variation of Stiffness. 283

25. Cases of Periodic Variation of Parametric Loads. 285

26. Pendulum with a Vibrating Point of Suspension. 288

27. Cases of Periodic Variation of the Inertia of a System. 289

**CHAPTER VI. SELF-EXCITED FRICTIONAL VIBRATIONS. 293**

28. Nature of Self-Excited Vibrations. 293

29. Self-Excited Vibrations of Quasi-Linear Systems. 298

30. Self-Excited Relaxation Vibrations. 304

Index. 314

]]>All credits to The Internet Archive user @librosmir

Curso De Algebra Superior A. Kurosch

Fundamentos Del Análisis Matemático 1 V. Ilín & E. Pozniak

Fundamentos Del Análisis Matemático 2 V. Ilín & E. Pozniak

Fundamentos Del Análisis Matemático 3 V. Ilín & E. Pozniak

Geometría Elemental A. Pogorélov

LPM ( 08) Resolución De Ecuaciones En Números Enteros A. Guelfond

LPM Resolución De Ecuaciones En Números Enteros. A. O. Guelfond

Problemas De Geometría Analítica D. Kletenik

Problemas De Geometría Descriptiva V. Gordon, Y. Ivanov & T. Solntseva

Problemas De Geometría Diferencial A. Fedenko

Geometría Moderna B. Dubrovin, S. Nóvikov & A. Fomenko

Geometría Diferencial A. Pogorélov

Curso Breve De Geometría Analítica N. Efimov

]]>

**About the book**

The book is a theoretical treatise on the subject of higher order moments. The theory of moments developed into a specific branch of mathematics which became a useful tool for solving complicated problems in structural mechanics. A number of Soviet scientists developed the so-called moment-operational method, which has proved to be extremely efficient for solving various problems of modern engineering. These problems arise when non-linear elasticity must be taken into account, when precise designing of non-uniform structural elements is required, when the loading of a structure is essentially non-uniform, etc.

The moment-operational method has been widely employed for compiling manuals containing numerous tables and formulas for designing beams, arches and frames. A number of scientific research works and textbooks present the theory of moments of higher order methods examples of problems solved by the moment-operational method.

The knowledge of the theory of higher moments as well as of the moment-operational method will serve to extend the field of their application to new problems.

The book was translated from the Russian by *N. Lebedinsky* and was published by Mir in 1974.

Many thanks to *Akbar Azimi* for providing the raw scans. We did the cleaning from 2-in-1 scans. There might be warping in some pages but the text overall is very readable.

**Contents**

Part 1 Moments of Higher Order: Theory and Application 9

Chapter 1. Theory of Moments 9

1. General Concept of Moments of Area 9

2. Moments as Geometric Characteristics of Beam Cross Sections 18

3. Uniaxial Moments of Point Forces and Couples 36

4. Uniaxial Moments of Balanced and Unbalanced Systems 40

5. Uniaxial Moments of Areas of Simple Figures 45

6. Uniaxial Moments of Areas of More Complicated Figures 52

7. Uniaxial Moments of Compound Loads 56

8. Moments of Higher Order and Generalized Forces 64

Chapter II. Application of Theory of Moments 73

9. Rigidity or Uniform Beams 73

10. Geometrical Interpretation of Moments 74

11. Calculation of Displacement Integrals in Rod Systems 83

12. Application of Higher Moments to Loading of Parabolic Influence Lines 92

13. Formulas for Statically Indeterminate Structures 94

14. Rigidity of Beams Composed of Prismatic Parts 140

15. Moments of Area of Flexibility Diagram for Non-Uniform Beams 145

Part II Moment-Operational Method: Theory and Application 160

Chapter III. Moment-Operational Method 160

16. Principles of Moment-Operational Method 160

17. Bimoments 162

18. Differential and Integral Bimoments 163

19. Application of Moment-Operational Method for Solving

Linear Differential Equations 170

Chapter IV. Rigidity of Non-Uniform Beams 173

20. Determination of Displacements by Coefficients of Flexibility Polynomial Expression 173

21. Determination of Displacements by Derivatives of Flexibility Analytical Expression 178

22. Determination of Displacements by Flexibility Integrals 182

23. Dermination of Displacements when Rigidity Follows Power 194

24. Determination of Displacement Integrals by Coefficients of Flexibility Polynomial 196

25. Determination of Displacement Integrals by Derivatives of Flexibility Analytical Expression 199

Chapter V. Multispan Non-Uniform Beams 202

28. Equation of Three Moments in Flexibility Polynomial Coefficients 202

27. Equation of Three Moments in Derivatives of Analytically Expressed Flexibility 207

28. Equation of Three Moments in Flexibility Integrals 210

29. Mohrs Integrals for Non-Uniform Beams 219

Chapter VI. Beams on Elastic Foundation 223

30. General 223

31. Prismatic Beams on Foundation of Constant Rigidity 224

32. Prismatic Beams on Foundation of Linear Rigidity 227

33. Beams on Foundation of Hyperbolic Rigidity 229

34. Beams on Elastic Foundation with Moment Reaction 236

Chapter VII. Beams Under Combined Flexure and Compression 238

35. General 238

36. Prismatic Beams Under Arbitrary Transverse Loads and Constant Axial Forces 239

37. Rotating Rod of Constant Rigidity Under Compression and Flexure 255

38. Prismatic Beam Under Arbitrary Transverse Load and Uniformly Distributed Axial Forces 260

39. Prismatic Beam Under Arbitrary Transverse Load and Linear Axial Forces 270

40. Prismatic Beam Under Arbitrary Transverse Load and Axial Law Distributed Along the Beam According to Polynomial 283

Chapter VIII. Application of Moment-Operational Method to Certain Complex Problems 285

41. Stability of Bars Under Axial Compression 285

42. Beams on Elastic Foundation Under Combined Compression and Bending 290

43. Non-Uniform Beam Under Axial Force 294

44. Higher Moments of Vector Quantities in Space 306

45. Rigidity of Beams: General Case of Non-Linear Stress-Strain Relationship 320

Chapter IX. Application of Moment-Operational Method to Structural Mechanics of Ships 331

46. Flexure of Irregular Decks 331

47. Non-Uniform Beams on Elastic Foundation 341

48. Stability and Vibration of Irregular Decks 346

This book holds a special place for me, as this book helped me understand many subtle points in physics. Also, this was the first book that I have ever scanned and had added to gigapedia when it was extant. Also, this was one of the first books that I took to typeset in LaTeX, almost a decade back, but gave up after many attempts and the project was untouched for several years. At end of 2019 I restarted the work and here we are.

One major challenge remains to convert the 130 odd diagrams to purely LaTeX using TikZ. I have done this for some figures (~6-7), but most of them are from the scans. I will do it as and when time allows. And of course you are free to contribute as well. Any help in this regard would be highly appreciated.

I have done a round of copy-editing, but still minor typos may exist here and there, (hopefully there are no major typos or screw-ups). So do report if you find any. Earlier scan had two pages which were missing (pages 36-37), they have been added in the current version, so this version is complete. We have also re-scanned the full page figures of section heads and coloured them. Many thanks to *psmitak* for the scans!

**About the typesetting **

The typesetting was fun, and I am pleased with the results. A lot of help was derived from questions on tex.stackexchange.com.

The main font used isURW-Garamond with mathdesign, while the sans font is TeX Gyre Adventor. The template used for the typesetting is tufte-book and the paper size is b5. The colour scheme used is `Maroon`

and `SteelBlue`

from `svgnames`

of `xcolor`

package of LaTeX along with DarkGray. I have, at times, highlighted questions using Maroon and answers or pedagogically significant remarks using SteelBlue. This is mostly done, if not completely done. Hope that this typeset version is helpful!

**PS:** Next in line is Tarasov’s Basic Concepts of Quantum Mechanics – one round is done, final copy-editing is currently in progress. A few glimpses from that project:

About the book

The book presents a systematic and step-by-step approach to the physical and mechanical properties of concrete, reinforcing steel, reinforced concrete, masonry, and structural steel.

The theoretical basis and design principles for reinforced concrete, masonry, and steel structures have been brought in line with new standards which were put in force in the Soviet Union since January, 1977. The material is presented with emphasis on Soviet, practice in highly industrialized precast reinforced concrete, type-design structural elements, and high-strength materials.

A good deal of attention is given to special engineering structures for water supply, sewage disposal, and heat supply systems, including tanks, settlers, aeration filters, aeration tanks, water lowers, buried conduits and headers, and heat-pipeline structures.

Carefully worked design examples are shown throughout the book, and an appendix gives a selection of design charts associated with the basic Soviet codes of practice.

Intended as textbook for students of building, civil, and structural engineering, the book will also be of interest to students in other departments and practising engineers.

The book was published by Mir in 1980 and was from the Russian by Alexander Kuznetsov.

Many thanks to *Akbar Azimi* for providing the original scans. We cleaned the 2-in-1 scan and created the pdf. There might be some warping in few of the pages, but overall it is very readable.

CONTENTS

Preface 8

Chapter One. Basics of Structural Design. 9

I.1. Limit States. 9

I.2. Design Factors. 10

I.3. Characteristic and Design Loads and Strength. 11

I.4. Limit-State Design. 13

I.5. Units of Measure. 15

Chapter Two. Reinforced Concrete as a Construction Material. 16

II.1. General. 16

II.2. Concrete. 19

II.3. Reinforcing Steel. 25

II.4. Main Properties of Reinforced Concrete. 35

Chapter Three. Analysis and Design of Reinforced Concrete Members. 37

III.1. General. 37

III.2. Design of Prestressed Concrete Structures. 40

III.3. Practical Hints for Design and Engineering. 47

III.4. Engineering Hints for Prestressed Members. 51

Chapter Four. Members in Axial Compression. 55

IV.1. Constructional Features. 55

IV.2. Design Under Accidental Eccentricity. 57

Chapter Five. Reinforced Concrete Members in Axial Tension 62

V.l. Constructional Features. 62

V.2. State of Stress and Design of Nonprestressed Members. 63

V.3. State of Stress and Design of Prestressed Members. 67

Chapter Six. Reinforced Concrete Members in Bending. 72

VI. 1. Constructional Features. 72

VI.2. Stale of Stress in Bending Members. 80

VI.3. Normal-Section Strength Analysis of Members. 85

VI.4. Inclined-Section Shear Strength Analysis. 98

VI.5. Incipient-Cracking Resistance of Prestressed Members. 108

VI.6. Sag Analysis. 114

VI.7. Crack-Opening Analysis. 118

Chapter Seven. Reinforced’ Concrete Members in Eccentrical Compression and Tension. 120

VII.1. Constructional Features of Members in Eccentrical Compression. 120

VI1.2. Design of Members in Eccentrical Compression. 123

VII.3. Members in Eccentrical Tension. 132

Chapter Eight. Masonry and Reinforced Masonry Structures. 135

VIII.1. Masonry Materials and Strength. 135

VIII.2. Design of Masonry Members in Compression. 138

VIII.3. Reinforced Masonry Structures. 142

VIII.4. Design of Masonry Structures. 145

VIII.5. Worked Examples for Design of Masonry Members. 147

Chapter Nine. Metal Structures. 152

IX.1. Materials for Metal Structures. 152

IX.2. Joints in Metal Structures. 157

IX.3. Design and Proportioning of Beams. 164

IX.4. Design and Proportioning of Columns. 175

IX.5. Design and Proportioning of Trusses. 188

IX.6. Prestressed Steel Structures. 191

Chapter Ten. Design of Buildings. 193

X.l. Principles of Building Layout. 193

X.2. Reinforced Concrete Floors. 196

X.3. Reinforced Concrete Column Footings. 219

X.4. Prefab Reinforced Concrete One-Storey Industrial Buildings. 227

X.5. Buildings for Water Supply and Sewage Disposal Systems and Boilers. 237

Chapter Eleven. Special Structures for Water Supply and Sewage Disposal Systems. 246

XI.1. General. 246

XI.2. Construction Types of Circular Reinforced Concrete Tanks. 249

XI.3. Design of Circular Tanks. 260

XI.4. Construction Types of Reinforced Concrete Rectangular Tanks. 273

XI.5. Design of Rectangular Tanks. 285

XI.6. Construction Types and Design of Steel Circular Tanks. 292

XI.7. Reinforced Concrete Pipes and Wells for Water Supply and Sewage Disposal Systems. 296

XI.8. Construction Types and Design of Water Towers. 300

Chapter Twelve. Design of Reinforced Concrete Structures. Worked Examples. 306

XII.l. General. 306

XII.2. Nonprestrossod Ribbed Roof Slab. 306

XII.3. Prestressed Ribbed Roof Slab. 317

XII.4. Squaro Roof Slab. 324

XII.5. Roof Girder. 329

XII.6. Column and Footing. 337

XII.7. Wall of a Precast Reinforced Concrete Rectangular Tank. 342

XII.8. Wall of a Precast Circular Tank. 348

Chapter Thirteen. Structures for Heat Supply Systems. 355

XIII.1. General. 355

XIII.2. Construction Types of Conduits and Headers. 357

XIII.3. Servicing Chambers, Compensating Niches, and Supports for Heat Pipelines. 367

XIII.4. Heat Pipelines Laid without Conduits. 378

XIII.5. Overhead Heat Pipelines. 379

XIII.6. Design Principles for Heat Pipeline Structures.386

Appendices. 395

Index. 413

]]>The book has come out nicely. Of course, it could have been better! I have typeset it in A5 paper, with 12 pt font with LuaLaTeX. The font used is EB Garamond. I have tried to maintain the typesetting of the original book with no chapter numbers or section numbers.

The table of elementary particles and their properties was one of the more challenging tasks in this otherwise simple book to typeset. It took me almost a day to just typeset this table. The rest was easy.

I have also created the front and back covers using TikZ, the first time I have tried this. Though they could have been done better.

The source files can be found at the gitlab project page:

https://gitlab.com/mirtitles/rydnik-abc-qm

**Note:** The original scan (by itanveer) which we had cleaned and posted had two pages (258-259) missing. I had already made a draft post asking for help for these two pages from people who might have the book, as I do not currently have the access to my physical collection. Except for these two pages, the rest of the book was processed and ready. Serendipitously, just a couple of days back **Hemant Garach** commented on the earlier post saying that he has this book! I requested him to scan these two missing pages and thanks to him we now have the complete book!

**TODO:** What remains to be done (in order of need):

- One more round of copy-editing. Though I tried my best, there will be many small typos here and there. I hope I have not screwed up in any major way. A thousand eyes are better than just two!
- Recreating figures (and front cover) using TikZ or other tools, so that it becomes completely electronic. For this release, I have used the scanned raster figures. Some actual photos of cloud chambers and particle tracks can be used instead of the current ones.
- There has been much increase in our knowledge of particle physics (both empirical and theoretical) ever since the book was written in the 1960s. The book itself might be expanded as much of the information might seem dated to a particle physicist. Perhaps a new expanded edition of the book covering the era from the 1970s to present? Anyone interested?!

The collection is a detailed selection of problems on the dynamics of the motion of a material point acted on by a central gravitational force, in particular, the dynamics of space flight. As an exception, the book presents several problems on the motion of a point acted on central non-gravitational forces. The book is written mainly for correspondence students. Topics covered include Kepler’s laws, the integral of areas, Binet’s formulas for central forces, the energy balance and velocity along a space trajectory, time of motion along a space trajectory, conditions for the existence of elliptical trajectories, transfer from orbit to orbit, sphere of action, third escape velocity problems, two-body problem, and the generalized third law of Kepler, along with miscellaneous problems.

This collection is a textbook for the course of theoretical mechanics (“Point Dynamics” section). students in correspondence departments of Leningrad State University and other higher educational institutions used in part by students in the day and evening departments. Moreover, the problem book may prove useful for beginning instructors in providing practical exercises mechanics, particularly, when they prepare modifications of test problems.

Most of the collection is a detailed sampling of problems on the dynamics of a material point acted on by gravitational force, in particular, problems on the elementary dynamics of space flight. Several problems on the motion of a point acted on by central non-gravitational forces are presented. Altogether, the collection includes about 200 problems of varying degrees of difficulties, with solutions.

Translation of *Sbornik zadach po dinamike tochki v pole tsentral’nykh sil,* Leningrad, Leningrad University Press, 1974, pp. 1-145. The book was published by NASA in May 1975 under its technical translation programme (NASA TT F-16,263).

Foreword iv

Table of Contents vi

Chapter One. Central Forces. Force of Gravity and Its Dynamic Characteristics 1

Chapter Two. Kepler’s Laws 11

Chapter Three. Integral of Areas 19

Chapter Four. Binet’s Formulas for Central Forces 26

Chapter Five. Energy Balance and Velocity Along a Space Trajectory 36

Chapter Six. Time of Motion in a Space Trajectory 63

Chapter Seven. Conditions for the Existence of Elliptical Trajectories 77

Chapter Eight. Transfer from Orbit to Orbit 90

Chapter Nine. Sphere of Action. Problems of Third Escape Velocity 109

Chapter Ten. Two-Body Problem. Third Kepler’s Law Generalized 118

Chapter Eleven. Miscellaneous Problems 127

References 163

N. V. Efimov, E. R. Rozendorn

About the book

This book was conceived as a text combining the course of linear algebra and analytic geometry. It originated as a course of lectures delivered by N. V. Efimov at Moscow State University (mechanics and mathematics department) in 1964-1966. However, the material of these lectures has been completely reworked and substantially expanded. We have tried to bear in mind the requirements of other mathematical disciplines and also of mechanics and physics. We hope that all parts of the text will be useful. The only preparation required for this text can be given an a first- semester course of analytic geometry and algebra at the most elementary level. All that is needed is a firm grasp of the elements of these subjects. For Chapter XII the student should be acquainted with projective transformations and the projective properties of figures in the plane. Also, in Chapter X the reader may simplify his task by skipping Subsections 13 to 23 (Section 3) and Subsection 10 of Section 7. What is left of Chapter X can serve as a minimal algebraic basis for the theory of multidimensional integration.

It may be noted in conclusion that the first five chapters already contain material with broad applications in mathematics, mechanics, and physics. These chapters, supplemented with some of the material of subsequent chapters, can be utilized in higher technical schools with a more advanced mathematics curriculum.

The book was translated from the Russian by George Yankovsky and published by Mir in 1975.

Many thanks to shankar.leo for providing the scans and the pdf.

Note: I tried to optimise the file for size, but somehow the archive kept on rejecting for some error in the pdf. I could have tried a few more things, but didn’t get time for that, hence the delay in post. For now, hence this large file (~120M) was uploaded. This file is OCRed but without bookmarks and pagination.

I will try to update a smaller file in the future, or if someone can add a link to a smaller file, it would be great.

Preface 9

Introduction 11

Chapter I. Linear Spaces

1. Axioms in linear space 15

2. Examples Of linear spaces 17

3. Elementary corrolaries to the axioms of a linear space 23

4. Linear combinations. Linear dependence.25

5. Lemma on the basis minor 27

6. Basic lemma on two systems of vectors 30

7. The rank of a matrix 32

8. Finite-dimensional and infinite-dimensional spaces. Bases 34

9. Linear operations in components 36

10. Isomorphism between linear spaces 38

11. Correspondence between complex and real spaces 40

12. Linear subspace 42

13. Linear hull 44

14. Sum of subspaces. Direct sum 47

Chapter II. Linear Transformations of Variables. Transformations of Coordinates

1. Abbreviated notation for summation 53

2. Linear transformation of variables. The product of linear

transformations of variables and matrix products 56

3. Square matrices and nonsingular transformations 60

4. The rank of a product of matrices 64

5. Transformation of coordinates in a change of basis 66

Chapter III. Systems of Linear Equations. Planes In Affine Space

1. Affine space 70

2. Affine coordinates 71

3. Planes 73

4. Systems of first-degree equations 77

5. Homogeneous systems 81

6. Nonhomogeneous systems 88

7. Mutual positions of planes 91

8. Systems of linear inequalities and convex polyhedrons 98

Chapter IV. Linear, Bilinear and Quadratic Forms

1. Linear forms 108

2. Bilinear forms 112

3. The matrix of a bilinear form 116

4. Quadratic forms 118

5. Reducing a quadratic form to canonical form by Lagrange’s method 121

6. The normal form of a quadratic form 124

7. The law of inertia of quadratic forms 125

8. Reducing a quadratic form to canonical form by Jacobi’s method 127

9. Positive definite and negative definite quadratic forms 129

10. Gram’s determinant. The Cauchy-Bunyakovsky inequality 132

11. Zero subspaces of a bilinear and a quadratic form 134

12. The zero cone of a quadratic form.137

13. Elementary examples of zero cones of quadratic forms 139

Chapter V. Tensor Algebra

1. Reciprocal bases. Contravariant and covariant vectors 142

2. Tensor product of linear spaces 149

3. Basis in a tensor product. Components of a tensor 153

4. Tensors of bilinear forms 159

5. Multiple-order tensors. Tensor product 162

6. Components of multiple-order tensors 166

7. Multilinear forms and their tensors 168

8. Symmetrization and antisymmetrization (alternation). Skewsymmetric forms 170

9. An alternative description of the tensor product of two linear spaces 174

Chapter VI. Groups and Some Applications

1. Groups and subgroups. Distribution of bases into classes

with respect to a given subgroup of matrices. Orientation 180

2. Transformation groups. Isomorphism and homomorphism of groups 186

3. Invariants. Axial invariants. Pseudoinvariants 191

4. Tensor quantities 197

5. The oriented volume of a parallelepiped. The discriminant tensor 201

Chapter VII. Linear Transformations of Linear Spaces

1. Generalities 207

2. A linear transformation as a tensor 210

3. The geometrical meaning of the rank and determinant of a linear transformation. The group of nonsingular linear transformations.213

4. Invariant subspaces 216

5. Examples of linear transformations 218

6. Eigenvectors and the characteristic polynomial of a transformation 224

7. Basic theorems on the characteristic polynomial and eigenvectors 227

8 Nilpolent transformations. The general structure of singular

transformations 229

9. The canonical basis of a nilpotent transformation 233

10. Reducing a transformation matrix to the Jordan normal form 242

11. Transformations of a simple structure 248

12. Equivalence of matrices 250

13. The Hamilton-Cayley formula 252

Chapter VIII. Spaces with Quadratic Metric

1. Scalar products 254

2. The norm of a vector 256

3. Orthonormal bases 258

4. Orthogonal projection. Orthogonalization 259

5. Metric isomorphism 265

6. ^-orthogonal matrices and ^ orthogonal groups 266

7. The group of Euclidean rotations 270

8. The group of hyperbolic rotations 278

9. Tensor algebra in quadratic-metric spaces 287

10. The equation of a hyperplane in quadratic-metric space 295

11. Euclidean space. Orthogonal matrices. Orthogonal group 297

12. The normal equation of a hyperplane in Euclidean space 302

13. The volume of a parallelepiped in Euclidean space. The discriminant tensor. Vector product 304

Chapter IX. Linear Transformations of Euclidean Space

1. Adjoint of a transformation 308

2. Lemma on the characteristic roots of a symmetric matrix 310

3. Self-adjoint transformations 311

4. Reducing a quadratic form to canonical form in an orthonormal basis 317

5. The joint reduction to canonical form of two quadratic forms 319

6. Skew-adjoint transformations 322

7. Isometric transformations 325

8. The canonical form of an isometric transformation 330

9. The motion of a rigid body with one fixed point 335

10. The curvature and torsion of a space curve 338

11. The decomposition of an arbitrary linear transformation into the product of a self-adjoint and an isometric transformation 340

12. Applications to the theory of elasticity. The strain tensor and the stress tensor 343

Chapter X. Multivectors and Outer Forms

1. Alternation 346

2. Multivectors. Outer product 351

3. Bivectors 357

4. Simple multivectors 366

5. Vector product 370

6. Outer forms and operations on them 376

7. Outer forms and covariant multivectors 379

8. Outer forms in three-dimensional Euclidean space 386

Chapter XI. Quadric Hypersurfaces

1. The general equation of a quadric hypersurface 391

2. Changes in the left member of the equation under translation of the origin 392

3. Changes in the left member of the equation for a change in the orthonormal basis 395

4. The centre of a quadric hypersurface 397

5. Reducing to canonical form the general equation of a quadric hypersurface in Euclidean space 399

6. Classification of quadric hypersurfaces in Euclidean space 402

7. Affine transformations. 410

8. Affine classification of quadric hypersurfaces 414

9. The intersection of a straight line with a quadric hypersurface. Asymptotic directions 415

10. Conjugate directions.418

Chapter XII. Projective Space

1. Homogeneous coordinates in affine space. Points at infinity 422

2. The concept of a projective space 425

3. A bundle of planes in affine space 435

4. Central projection 443

5. Projective equivalence of figures 446

6. Projective classification of quadric hypersurfaces 453

7. The intersection of a quadric hypersurface and a straight line. Polars 459

Appendix 1. Proof of the theorem on the classification of linear quantities 467

Appendix 2. Hermitian forms. Unitary space.471

Bibliography. 484

Index. 486

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