Mathematical Analysis – A Brief Course For Engineering Students – Bermant, Aramanovich

Posted on May 4, 2022 by The Mitr

In this post, we will see the book Mathematical Analysis – A Brief Course For Engineering Students by A.F . Bermant; I. G. Aramanovich.

About the book

This course is designed as a textbook for engineering students. It embraces the topics in mathematical analysis usually included into curricula of technical colleges. The course also contains some optional material which may be omitted in a first reading of the book; the corresponding items are marked with the asterisk.
There are a number of courses dealing with special divisions of mathematical analysis, such as equations of mathematical physics, functions of a complex argument and the like, and therefore, although these divisions are important for mathematical education of an engineer, they are not treated in this book. We also draw attention to the fact that only a few questions related to approximate calculations and programming (e.g. the applica­tion of the differential to approximate calculations, methods of approximate solution of equations, numerical integration and solution of differential equations, etc.) are discussed in this course. For a thorough study of this subject some other textbooks should be used.
In this course many examples are given which demonstrate the application of mathematical analysis to various divisions of mechanics and physics. The study of these examples is very impor­tant since the main interest of an engineer lies in solving concrete applied problems. At the end of each chapter we give a number of questions aimed at checking the understanding of the theore­tical material. In the presentation of the material the main emphasis has been laid upon practical aspects, and some purely mathematical facts are given without proof. On the other hand, in some cases detailed proofs of theorems are given, especially when this elucidates the meaning of the theorem and shows in which way it can be applied. Besides, the study of the proofs helps the student to acquire practice in logical argument and provides prerequisites for further mathematical self-education.

The book was translated from Russian by was published in 1986  by Mir Publishers.

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Contents

(The starred items indicate the sections that may
be omitted in a first reading of the book)

Preface 5
Introduction 15

1. The Subject of Mathematical Analysis 15
2. Variables and Functions 15
3. The Role of Mathematics and Mathematical Analysis in Natural Sciences and Engineering 16

CHAPTER I. FUNCTION 19

§ 1. Real Numbers 19
§ 2. The Concept of Function 26
§ 3. Characteristics of Behaviour of Functions. Some Important Examples 38
§ 4. Inverse Function. Power, Exponential and Logarithmic Function 51
§ 5. Trigonometric, Inverse Trigonometric, Hyperbolic and Inverse
Hyperbolic Functions. 60

CHAPTER II. LIMIT. CONTINUITY 72

§ 1. Limit. Infinitely Large Magnitudes 72
§ 2. Continuous Functions 98
Oo. CONUIAUIN ai ads Sa aca & ee Ss ae ee eee ee :
§ 3. Comparison of Infinitesimals. Comparison of Infinitely Large
Magnitudes 109

CHAPTER III. DERIVATIVE AND DIFFERENTIAL. DIFFERENTIAL CALCULUS 118

§ 1. Derivative 118
§ 2. Differentiating Functions 126
§ 3. Some Geometrical Problems. Graphical Differentiation 148
§ 4. Differential 155
§ 5. Derivatives and Differentials of Higher Orders 167

CHAPTER IV. APPLICATION OF DIFFERENTIAL CALCULUS TO INVESTIGATION OF BEHAVIOUR OF FUNCTIONS 175

§ 1. Theorems of Fermat, Rolle, Lagrange and Cauchy 175
§ 2. Investigating Functions wita the Aid of First and Second Derivatives 181
§ 3. L’Hospital’s Rule. General Scheme for Investigating Functions 204
§ 4. Curvature 219
§ 5. Space Curves. Vector Function of a Scalar Argument 225
§ 6. Complex Functions of a Real Argument 237
§ 7. Solution of Equations 245
QUESTIONS 255

CHAPTER V. INTEGRAL CALCULUS 258

§ 1. Indefinite Integral 258
§ 2. Definite Integral 291
§ 3. Methods of Evaluating Definite Integrals. 318
§ 4. Improper Integrals 331

CHAPTER VI. APPLICATION OF INTEGRAL CALCULUS 345

§ 1. Some Problems of Geometry and Statics 345
§ 2. General Scheme of the Application of the Integral 358

CHAPTER VII. FUNCTIONS OF SEVERAL VARIABLES AND THEIR DIFFERENTIATION 365

§ 1. Functions of Several Variables 365
§ 2. Derivatives and Differentials. Differential Calculus 376
§ 3. Applications of Differential Calculus to Geometry 409
§ 4. Extrema of Functions of Two Variables 414
§ 5. Scalar Field 427

CHAPTER VIII. DOUBLE AND TRIPLE INTEGRAL 437

§ 1. Double Integrals 437
§ 2. Triple Integrals 459
§ 3. Integrals Dependent on Parameters 471

CHAPTER IX. LINE INTEGRALS AND SURFACE INTEGRALS. FIELD THEORY 482

§ 1. Line Integrals 482
§ 2. Surface Integrals 515
§ 3. Field Theory 533
Questions 561

CHAPTER X. DIFFERENTIAL EQUATIONS 564

§ 1. Differential Equations of the First Order 564
§ 2. Differential Equations of the Second and Higher Orders 592
§ 3. Linear Differential Equations 603
§ 4. Systems of Differential Equations 635
QUESTIONS 656

CHAPTER XI. SERIES 659

§ 1. Numerical Series 659
§ 2. Functional Series 678
§ 3. Power Series 684
§ 4. Expanding Functions into Power Series 691
§ 5. Some Applications of Taylor’s Series 706
§ 6*. Some Further Topics in the Theory of Power Series 716
Questions 721

CHAPTER XII. FOURIER SERIES AND FOURIER INTEGRAL. 724

§ 1. Fourier Series 724
§ 2. Some Further Topics in the Theory of Fourier Series 746
§ 3. Fourier-Integral 753
Questions 761

Table of Integrals 762

Bibliography 768

Name Index 770

Subject Index 772

 

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An Introduction To The Theory Of Plasma Turbulence – Tsytovich

Posted on May 3, 2022 by The Mitr

In this post, we will see the book An Introduction To The Theory Of Plasma Turbulence
by V. N. Tsytovich.

About the book

This book is based upon lectures given by Professor Tsytovich at Culham Laboratory. The preceding text can only represent the present state of the development of the theory of plasma turbulence. The author has tried to follow the logic, but not the history of this field and, therefore, the references are very fragmented and not by any means complete. The essential physical statements that the author wants to emphasise finally are:

  1. The plasma properties in the turbulent region are mostly non-linear. This raises the possibility of universal plasma properties like a universal spectrum that can be independent of the type of instability.

  2. Nevertheless, the turbulence is often weak: W/nT << 1, and when describing the properties of the turbulent oscillation interactions it is not possible to expand the non-linear interactions in terms of the turbulent energy. The elementary excitations such as plasmons and “dressed” particles have thus a finite lifetime which is connected with their non­-linear interactions.

  3. The small low-frequency perturbations in a turbulent plasma have quite a different nature because of the frequent turbulent collisions, and the dielectric constant that describes such perturbations cannot be expanded in terms of the turbulent energy.

  4. The development of a turbulent state is very probable for a plasma as a result of the fact that the energy applied has a tendency to disperse to the greatest possible degree of freedom. Innumerable numbers of different plasma instabilities can bring the plasma to a turbulent state. The plasmas in astrophysical conditions must, therefore, often be turbulent. This can lead to a way of explaining cosmic-ray origins with a universal power-type spectrum.

  5. The development of plasma turbulence can occur as a result of development, firstly, of one or a small number of collective modes, with a subsequent spread of the energy to other modes by non-linear inter­actions as well as by the excitation of many modes at the first stage. For the case of the excitation of one mode, the first stage is not turbulent and the turbulence develops as the energy is spread, if the system is ergodic. The plasma collective motions seem to be the best test for an investigation of the general problems of the development of the random­isation process, as well as of the general problems of the possibility of a statistical description of a system.

The book was translated from Russian and was published in  1972.

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Contents

 

1. Comparison of plasma and liquid turbulence 1
2. General Problems of the Theory of Plasma Turbulence 14
3. The Balance Equation for a Turbulent Plasma 27
4. Turbulent Collisions and Resonance Broadening 44
5. The Spectrum and Correlation Functions of Ion-sound Turbulence 62
6. The Spectrum and Correlation Functions of Langmuir Turbulence 74
7. Electromagnetic Properties of a Turbulent Plasma 93
8. The Cosmic-ray Spectrum 105

Conclusions 128

References 130

Index 133

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Method Of Edge Waves In The Physical Theory Of Diffraction – Ufimtsev

Posted on May 2, 2022 by The Mitr

In this post, we will see the book Method Of Edge Waves In The Physical Theory Of Diffraction by P. Ya. Ufimtsev.

About the book

The book is a monograph written as a result of research by the author. The diffraction of plane electromagnetic waves by ideally conducting bodies, the surface of which have discontinuities, is investigated in the book. The linear dimensions of the bodies are assumed to be large in comparison with the wavelength. The method developed in the book takes into account the perturbation of the field in the vicinity of the surface discontinuity and allows one to substantially refine the approximations of geometric and physical optics. Expressions are found for the fringing field in the distant zone. A numerical calculation is performed of the scattering characteristics, and a comparison is made with the results of rigorous theory and with experiments. The book is intended for physicists and radio engineers who are interested in diffraction phenomena, and also for students of advanced courses and aspirants who are specializing in antennas and the propagation of radio waves.

The book was translated from Russian and was published in 1962  by Foreign Technology Division of USA.

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Contents

FOREWORD

INTRODUCTION

CHAPTER I. DIFFRACTION BY A WEDGE 1

§ 1. The Rigorous Solution 1
§ 2. Asymptotic Expressions 12
§ 3. The Physical Optics Approach 18
§ 4. The Field Radiated by the Nonuniform part of the Current 26
§ 5. The Oblique Incidence of a Plane Wave on a Wedge 32
§ 6. Diffraction by a Strip 35

CHAPTER II. DIFFRACTION BY A DISK 43

§ 7. The Physical Optics Approach 43
§ 8. The Field from the Uniform Part of the Current 48
§ 9. The Total Field Being Scattered by a Disk with Normal Irradiation 52
§ 10. The Physical Optics Approach 54
§ 11. The Field Radiated the Nonuniform Part of the Current 57
§ 12. The Scattering Characteristics: with an Arbitrary Irradiation66

CHAPTER III. DIFFRACTION BY A FINITE LENGTH CYLINDER 73

§ 13. The Physical Optics Approach 74
§ 14. The Field Created by the Nonuniform Part of the Current 80
§ 15. The Total Fringing Field 83

CHAPTER IV. DIFFRACTION OF A PLANE WAVE INCIDENT ALONG THE SYMMETRY AXIS OF FINITE BODIES OF ROTATION 90

§ 16. The Field Created by the Nonuniform Part of the Current 90
§ 17. A Cone 95
§ 18. A Paraboloid of Rotation 103
§ 19. A Spherical Surface 108

CHAPTER V. SECONDARY DIFFRACTION 114

§ 20. Secondary Diffraction by a Strip. Formulation of the Problem 115
§ 21. Secondary Diffraction by a Strip (H-Polarization) 118
§ 22. Secondary Diffraction by a Strip (E-Polarization) 126
§ 23. The Scattering Characteristics of a Plane Wave by a Strip 129
§ 24. Secondary Diffraction by a Disk 138
§ 25. A Brief Review of the Literature 154

CHAPTER VI. CERTAIN PHENOMENA CONNECTED WITH THE NONUNIFORM PART OF THE SURFACE CURRENT 163

§ 26. Measurement of the Field Radiated by the Nonuniform part of the Current 163
§ 27. Reflected Wave Depolarization 170

CHAPTER VII. DIFFRACTION BY A THIN CYLINDRICAL CONDUCTOR 175

§ 28. Current Waves in an Ideally Conducting Vibrator 176
§ 29. Radiation of a Transmitting Vibrator 183
§ 30. Primary and Secondary Diffraction by a Passive Vibrator 185
§ 31. Multiple Diffraction of Edge Waves 193
§ 32. Total Fringing Field 196
§ 33. A Vibrator Which is Short in Comparison with the Wavelength (a Passive Dipole) 204
§ 34. The Results of Numerical Calculations 208

CONCLUSION 217
REFERENCES 221

 

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Diseases Of The Ear, Nose And Throat – Likhachev

Posted on May 1, 2022 by The Mitr

In this post, we will see the book Diseases Of The Ear, Nose And Throat by A. Likhachev.

About the book

The present text-book of otorhinolaryngology is intended for secondary medical schools and gives the most essential theoretical and practical information needed by the junior medical personnel engaged in independent practice.
It is calculated to enable the junior medical personnel, employed as assistant physicians in medic”al institutions or working on their own, to diagnose typical diseases of the ear, nose and throat, prescribe and give correct treatment, and if need he, render first aid to the patient.
Before discussing the clinical aspects of diseases of the ear, nose and throat, we deem it necessary to give a concise description of the anatomy and physiology of these organs, which should considerably facilitate the clinical study.
Special attention is devoted to early diagnosis of ear, nose and throat diseases, which is very important for both treatment and prophylaxis.

The book was translated from Russian (translators name is not given) was published in 1955 by Foreign Languages Publishing House.

Original scan from Digital Library of India.

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Contents

Preface 7

DISEASES OF THE EAR 11

Anatomy of the Ear 11
Physiology of the Ear 22
Examination of the Ear 27
General Methods of Nursing and Treatment of Ear diseases 37
Diseases of the External Ear 42
Inflammations of the Middle Ear
Intracranial Complications of Suppurative Otitis 83
Nonsuppurative Diseases of the Middle and Internal Ear 88
Deaf-Mutism 97
Traumatic Lesions of the Ear 99
Occupational Diseases of the Ear 102

DISEASES OF THE NOSE, PHARYNX AND LARYNX 106

Diseases of the Nose and Paranasal Sinuses 106

Anatomy of the Nose 106
Physiology of the Nose 111
General Methods of Treatment in Nasal Diseases 116
Diseases of the External Nose 120
Acute Inflammations of the Nose 133
Chronic Inflammations of the Nose (Chronic Rhinitis) 138
Vasomotor or Allergic Rhinitis 145
Disturbances of the Sense of Smell 147
Neoplasms in the Nose 148
Acute and Chronic Diseases of Paranasal Sinuses 150

Diseases of the Pharynx 161

Anatomy of the Pharynx 161
Physiology of the Pharynx 163
Methods of Examining the Pharynx 164
Acute Inflammations of the Pharynx 172
Chronic Inflammations of the Pharynx 194
Benign Tumours of the Pharynx 198
Malignant Tumours of the Pharynx 200

Diseases of the Larynx 202

Anatomy of the Larynx 202
Methods of Examining of the Larynx 205
General Methods of Treatment in Laryngeal Diseases 207
General Symptoms of Laryngeal Diseases 210
Acute Laryngitis 211
Chronic Laryngitis 213
Laryngeal Perichondritis 215
Benign Tumours of the Larynx 215
Malignant Tumours of the Larynx 217
Acute and Chronic Stenoses of the Larynx 218
Motor Disorders of the Larynx 221
Tuberculosis of the Upper Respiratory Tract 223
Syphilis of the Upper Respiratory Tract 227
Scleroma 232

Diseases of the Trachea 234

Anatomy of the Trachea 234
Tracheobronchoscopy 234
Intubation 237
Tracheotomy 238
Foreign Bodies in the Larynx, Trachea and Bronchi 245
Traumatic Lesions of the Upper Respiratory Tract 247
Occupational Diseases of the Upper Respiratory Tract 255

Diseases of the Esophagus 260

Anatomy of the Esophagus 260
Methods of Examining the Esophagus 260
Burns and Strictures of the Esophagus 261
Foreign Bodies of the Esophagus 262
Cancer of the Esophagus 263

Supplement: Health Education 265

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Ten Decimal Tables – Lyusternik (Ed.)

Posted on April 30, 2022 by The Mitr

In this post, we will see the book Ten Decimal Tables Of The Logarithms Of Complex Numbers And For The Transformation From Cartesian To Polar Coordinates
edited by L. A. Lyusternik.

About the book

The present tables were compiled in the Department for Approximate Computations of the Institute of Exact Mechanics and Computational Methods of the U.S.S.R. Academy of Sciences. The computations were carried out by this department in conjunction with the Computational- Experimental Laboratory of the Institute.

The book was translated from Russian by D. E. Brown and was published in 1965.

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Qualitative Methods In The Many Body Problem – Khilmi

Posted on April 29, 2022 by The Mitr

In this post, we will see the book Qualitative Methods In The Many Body Problem by G. F. Khilmi.

About the book

The n-body problem is the name usually given to the problem of the motion of a system of many particles attracting each other according to Newton’s law of gravitation. This is the classical problem of mathematical natural science, the significance of which goes far beyond the limits of its astronomical applications.

The n-body problem has been the main topic of celestial mechanics from the time of its inception as a science. Now that many of the problems of celestial mechanics have become part of geophysics, the central position of the w-body problem has been further strengthened.

The fundamental dynamical problem for a system of n gravitating bodies is the investigation and predetermina­tion of the changes in position and velocity that the particles undergo as the time varies. However, this is a very complex non-linear problem whose solution has not been possible under the present-day status of mathematical analysis.

In the first chapter, we give the equations and general integrals of the n-body problem, and we study the simplest theorems on the final motion due to Jacobi.

In the second chapter, we consider means of applying the method of dimensional analysis to the n-body problem. As far as we know, dimensional analysis has not been used before in the investigation of the final motion; though it does not yield definitive results in this area, it is very use­ ful in carrying out a preliminary analysis of the problem.

In the third chapter, we present our “method of continuous induction” and we consider some applications of it in which the final motion in the n-body problem is analyzed. This method allows us to obtain effective qualitative results, namely, it allows us to formulate suffi­cient conditions for the occurrence of certain types of final motions in the form of conditions on the initial state of the dynamical system.

The fourth chapter is devoted to the method of invariant measure. The application of this method to the many-body problem has necessitated working out a number of theorems on the measure theory of dynamical systems. These are presented at the beginning of the chapter. At the end of the chapter, we prove some very general theorems on the motion of a system of gravitating bodies using the method of invariant measure.

In the fifth chapter, an attempt is made to analyze some cases of the evolution of a system of n gravitating bodies on the basis of celestial mechanics. Here, we shall be concerned with the processes which are of cosmogonical interest, and which are accompanied by the conversion of mechanical energy into non-mechanical forms.

The book was translated from Russian by B. D. Seckler and was published in 1961.

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Contents

CHAPTER 1. Equations and general integrals of the n-body problem. Simplest theorems on the final motion 1

CHAPTER 2. Method of dimensional analysis 15

CHAPTER 3. Method of continuous induction 27

CHAPTER 4. Method of invariant measure 61

CHAPTER 5. Analysis of some cases of the evolution of a system of gravitating bodies 97

Bibliography 113
Index 117

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A Method For Studying Model Hamiltonians – A Minimax Principle For Problems In Statistical Physics – Bogolyubov Jr

Posted on April 28, 2022 by The Mitr

In this post, we will see the book A Method For Studying Model Hamiltonians – A Minimax Principle For Problems In Statistical Physics by N. N. Bogolyubov Jr..

About the book

In this book methods are proposed for solving certain problems in statistical physics which contain four-fermion interaction.

It has been possible, by means of “approximating (trial) Hamiltonians”, to distinguish a whole class of exactly soluble model systems. An essential difference between the two types of problem with positive and negative four-fermion interaction is discovered and examined. The determination of exact solutions for the free energies, single-time and many-time correlation functions, T-products and Green’s functions is treated for each type of problem.

The more general problem for which the Hamiltonian contains some terms with positive and others with negative four-fermion interaction is also investigated. On the basis of analysing and general­izing the results of Chapters 1 to 3, it becomes possible to formulate and develop a new principle, the minimax principle, for problems in statistical physics (Chapter 4).

The book was translated from Russian by P. J. Shepherd and was published in 1972.

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Contents

INTRODUCTION 1

§ I. GENERAL REMARKS 1
§ II. REMARKS ON QUASI AVERAGES 16

CHAPTER 1
PROOF OF THE ASYMPTOTIC RELATIONS FOR THE MANY-TIME CORRELATION FUNCTIONS

§ 1. GENERAL TREATMENT OF THE PROBLEM. SOME PRELIMINARY RESULTS AND FORMULATION OF THE PROBLEM 25

§ 2. EQUATIONS OF MOTION AND AUXILIARY OPERATOR INEQUALITIES 33

§ 3. ADDITIONAL INEQUALITIES 37

§ 4. BOUNDS FOR THE DIFFERENCE OF THE SINGLE-TIME AVERAGES 40

§ 5. REMARK (1) 47

§ 6. PROOF OF THE CLOSENESS OF AVERAGES CONSTRUCTED ON THE BASIS OF MODEL AND TRIAL HAMILTONIANS FOR “NORMAL” ORDERING OF THE OPERATORS IN THE AVERAGES 50

§ 7. PROOF OF THE CLOSENESS OF THE AVERAGES FOR ARBITRARY ORDERING OF THE OPERATORS IN THE AVERAGES 54

§ 8. ESTIMATES OF THE ASYMPTOTIC CLOSENESS OF THE MANY-TIME CORRELATION AVERAGES 57

CHAPTER 2
CONSTRUCTION OF A PROOF OF THE GENERALIZED ASYMPTOTIC RELATIONS FOR THE MANY-TIME CORRELATION AVERAGES 65

§ 1. SELECTION RULES AND CALCULATION OF THE AVERAGES 65

§ 2. GENERALIZED CONVERGENCE 70

§ 3. REMARK 74

§ 4. PROOF OF THE ASYMPTOTIC RELATIONS 76

§ 5. REMARK ON THE CONSTRUCTION OF UNIFORM BOUNDS 79

§ 6. GENERALIZED ASYMPTOTIC RELATIONS FOR THE GREEN’S FUNCTIONS 82

§ 7. THE EXISTENCE OF GENERALIZED LIMITS 85

CHAPTER 3
CORRELATION FUNCTIONS FOR SYSTEMS WITH FOUR-FERMION NEGATIVE INTERACTION 90

§ 1. CALCULATION OF THE FREE ENERGY FOR MODEL SYSTEMS WITH ATTRACTION 90

§ 2. FURTHER PROPERTIES OF THE EXPRESSIONS FOR THE FREE ENERGY 101

§ 3. CONSTRUCTION OF ASYMPTOTIC RELATIONS FOR THE FREE ENERGY 105

§ 4. ON THE UNIFORM CONVERGENCE WITH RESPECT TO 𝛳 OF THE FREE ENERGY FUNCTION AND ON BOUNDS FOR THE QUANTITIES 𝛿_{V} 111

§ 5. PROPERTIES OF PARTIAL DERIVATIVES OF THE FREE ENERGY FUNCTION. THEOREM 3.III 114

§ 6. RIDER TO THEOREM 3.III AND CONSTRUCTION OF AN AUXILIARY INEQUALITY 117

§ 7. ON THE DIFFICULTIES OF INTRODUCING QUASI-AVERAGES 120

§ 8. A NEW METHOD OF INTRODUCING QUASI-AVERAGES 124

§ 9. THE QUESTION OF THE CHOICE OF SIGN FOR THE SOURCE-TERMS 30

§ 10. THE CONSTRUCTION OF UPPER-BOUND INEQUALITIES IN THE CASE WHEN C = 0 131

CHAPTER 4
MODEL SYSTEMS WITH POSITIVE AND NEGATIVE INTERACTION COMPONENTS 137

§ 1. HAMILTONIAN WITH NEGATIVE COUPLING CONSTANTS (REPULSIVE INTERACTION) 137

§ 2. FEATURES OF THE ASYMPTOTIC RELATIONS FOR THE FREE ENERGIES IN THE CASE OF SYSTEMS WITH POSITIVE INTERACTION 141

§ 3. BOUNDS FOR THE FREE ENERGIES AND CORRELATION FUNCTIONS 143

§ 4. EXAMINATION OF AN AUXILIARY PROBLEM 146

§ 5. SOLUTION OF THE QUESTION OF UNIQUENESS 150

§ 6. HAMILTONIANS WITH COUPLING CONSTANTS OF DIFFERENT SIGNS. THE MINIMAX PRINCIPLE 154

REFERENCES 164

INDEX 169

 

 

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Calculus Of Variations – Gelfand, Fomin

Posted on April 27, 2022 by The Mitr

In this post, we will see the book Calculus Of Variations by I.M. Gelfand; S.V. Fomin.

About the book

This book is a modern introduction to the calculus of variations and certain of its ramifications, and I trust that its fresh and lively point of view will serve to make it a welcome addition to the English-language literature on the subject. The present edition is rather different from the Russian original. With the authors’ consent, I have given free rein to the tendency of any mathematically educated translator to assume the functions of annotator and stylist.

The problems appearing at the end of each of the eight chapters and two appendices were made specifically for the English edition, and many of them comment further on the corresponding parts of the text.

The book was translated from Russian by Richard Silverman and was published in 1963.

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Contents

1. ELEMENTS OF THE THEORY 1.

1: Functionals. Some Simple Variational Problems, 1.
2: Function Spaces, 4.
3: The Variation of a Functional. A Necessary Condition for an Extremum, 8.
4: The Simplest Variational Problem. Euler’s Equation, 14.
5: The Case of Several Variables, 22.
6: A Simple Variable End Point Problem, 25.
7: The Variational Derivative, 27.
8: Invariance of Euler’s Equation, 29. Problems, 31.

2. FURTHER GENERALIZATIONS 34.

9: The Fixed End Point Problem for n Unknown Functions, 34.
10. Variational Problems in Parametric Form, 38.
11: Functionals Depending on Higher-Order Derivatives, 40.
12: Variational Problems with Subsidiary Conditions, 42.
Problems, 50.

3. THE GENERAL VARIATION OF A FUNCTIONAL 54.

13: Derivation of the Basic Formula, 54.
14: End Points Lying on Two Given Curves or Surfaces, 59.
15: Broken Extremals. The Weierstrass-Erdmann Conditions, 61.
Problems, 63.

4. THE CANONICAL FORM OF THE EULER EQUATIONS AND RELATED TOPICS 67.

16: The Canonical Form of the Euler Equations, 67.
17: First Integrals of the Euler Equations, 70.
18: The Legendre Transformation, 71.
19: Canonical Transformations, 77. 20: Noether’s Theorem, 79.
21: The Principle of Least Action, 83.
22: Conservation Laws, 85.
23: The Hamilton-Jacobi Equation. Jacobi’s Theorem, 88.
Problems, 94,

5. THE SECOND VARIATION. SUFFICIENT CONDITIONS FOR A WEAK EXTREMUM 97.

24: Quadratic Functionals. The Second Variation of a Func-tional, 97.
25: The Formula for the Second Variation. Legendre’s Condition, 101. 26: Analysis of the Quadratic Functional | (Ph’? + Qh?) dx, 105.
27: Jacobi’s Necessary Condition. More on Conjugate Points, 111.
28: Sufficient Conditions for a Weak Extremum, 115.
29: Generalization to n Unknown Functions, 117.
30: Connection Between Jacobi’s Condition and the Theory of Quadratic Forms, 125.
Problems, 129.

6. FIELDS. SUFFICIENT CONDITIONS FOR A STRONG EXTREMUM 131.

31: Consistent Boundary Conditions. General Definition of a Field, 131.
32: The Field of a Functional, 137.
33: Hilbert’s Invariant Integral, 145.
34: The Weierstrass E-Function. Sufficient Conditions for a Strong Extremum, 146.
Problems, 150.

7. VARIATIONAL PROBLEMS INVOLVING MULTIPLE INTEGRALS 152.

35: Variation of a Functional Defined on a Fixed Region, 152.
36: Variational Derivation of the Equations of Motion of Continuous Mechanical Systems, 154.
37: Variation of a Functional
Defined on a Variable Region, 168.
38: Applications to Field Theory, 180.
Problems, 190.

8. DIRECT METHODS IN THE CALCULUS OF VARIATIONS 192.

39: Minimizing Sequences, 193.
40: The Ritz Method and the Method of Finite Differences, 195.
41: The Sturm-Liouville Problem, 198.
Problems, 206.

APPENDIX I PROPAGATION OF DISTURBANCES AND THE CANONICAL EQUATIONS 208.

APPENDIX II VARIATIONAL METHODS IN PROBLEMS OF OPTIMAL CONTROL 218.

BIBLIOGRAPHY 227.

INDEX 228.

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Forging Practice – Kamenshchikov, Koltun, Naumov, Chernobrovkin

Posted on April 26, 2022 by The Mitr

In this post, we will see the book Forging Practice by G. Kamenshchikov; S. Koltun; V. Naumov; B. Chernobrovkin.

About the book

All machines are built up of parts made of different materials and by various manufacturing processes. Some parts are cast from metals; some are forged, while others are produced by machining on different kinds of machine tools. Castings and forgings have to be machined before they acquire their proper shape, exact dimensions and surface finish. Forged parts, whether they are to be machined or not, are called forgings. This book describes various technologies used in the forgings.

The book was translated from Russian (translator’s name is not mentioned) was published  by Peace Publishers in 1960.

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Contents

Introduction 7
The Importance of Forging in Machine Building7

Chapter I. Bench Operations 8
Chapter II. Introduction to Forging Practice 18
Chapter III. Fuel and Its Combustion 36
Chapter IV. Heating Devices. 60
Chapter V. Heating Steel for Forging 101
Chapter VI. Chief Hand-Forging Operations 115
Chapter VII. The Influence of Deformation on Forgings and the Calculation of Forgings 161
Chapter VIII. Hammers for Hammer Forging 190
Chapter IX. Forging Operations and Hammer Forging Tools 232
Chapter X. The Technological Process and Examples of Hammer Forging 259
Chapter XI. Forging presses and Their Operation 278
Chapter XII. Automatic Forging and Stamping Machines 313
Chapter XIII. Drop-Forging (Hot Stamping) 322
Chapter XIV. Special Features of Forging Non-Ferrous Metals and Their Alloys 366
Chapter XV. Heat Treatment, Defects and Inspection of Forgings 368
Chapter XVI. Organisation of Work and of the Working Place 376
Chapter XVII. Safety Engineering 398

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Quantum Mechanics – Davydov

Posted on April 25, 2022 by The Mitr

In this post, we will see the book Quantum Mechanics by A. S. Davydov.

About the book

The present book is an extended exposition of a course of lectures in quantum mechanics given by the author over several years to students of the Physics Depart­ment of Moscow University. This course is given after the sections on atomic and nuclear physics of the general course in physics. In those sections of the course a historical outline of the development of contemporary ideas about the structure of atoms and of atomic nuclei is given, as well as the experimental data on which quantum mechanics is based. The present book therefore does not touch at all upon the historical development of quantum theory.
The main emphasis in the present book is upon the physical ideas and the mathematical formalism of the quantum theory of the non-relativistic and quasi-relativistic (up to terms of order v2lc2) motion of a single particle in an external field. In particular, we show the inapplicability of the concept of an essentially relativistic motion of a single particle. We put great emphasis upon representation theory, the theory of canonical transformations, scattering theory, and quantum transitions. A relatively detailed exposition is given of the theory of systems consisting of identical bosons or fermions. We also devote several sections to the theory of molecules, the theory of chemical binding, and solid state theory.
An important role is played in this book by the theory of second quantisation as a method to study systems consisting of a large number of identical particles. In parti­cular, we give the basic ideas of the theories of superconductivity and of superfluidity. The basic ideas are given of the methods for quantising the meson field, the electro­magnetic field (without charges) and the electron-positron field, neglecting diver­gencies and renormalisation, as these topics are dealt with in special books which are studied after quantum mechanics.
The present book can be used as an introduction to a study of quantum elctrodynamics, nuclear theory, or solid state theory. To read it, it is necessary to be fami­liar with the usual contents of university courses in mathematics, classical mechanics, and electrodynamics. For reference purposes we give at the end of the book some mathematical appendices about special functions, matrices, and group theory.
In this book we mainly refer to review or original papers when we want to indicate where the reader can study a more detailed discussion of a topic. These references do not pretend to be complete.
Although we do not consider in this book special methodological problems, the exposition is based upon dialectic materialism, that is, we start from the idea that the regularities of atomic and nuclear physics which are studied in quantum mechanics are objective regularities of nature.

The book is intended for students of physics, studying quantum mechanics. It can also be used as a reference book for teachers and other scientists.

The book was translated from Russian by D. Ter Haar was published in 1965.

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Contents

PREFACE

PREFACE TO THE ENGLISH EDITION

CHAPTER I. THE BASIC CONCEPTS OF QUANTUM MECHANICS 1
CHAPTER II. CHANGE OF QUANTUM STATES IN TIME 45
CHAPTER III. THE CONNEXION BETWEEN QUANTUM MECHANICS AND CLASSICAL MECHANICS 69
CHAPTER IV. ELEMENTARY REPRESENTATION THEORY 85
CHAPTER V. THE SIMPLEST APPLICATIONS OF QUANTUM MECHANICS 106
CHAPTER VI. THE MOTION OF A PARTICLE IN A CENTRAL FIELD OF FORCE 123
CHAPTER VII. APPROXIMATE METHODs FOR EVALUATING EIGENVALUES AND EIGENFUNCTIONS 169
CHAPTER VIII. THE FOUNDATIONS OF A QUASI-RELATIVISTIC QUANTUM THEORY OF THE MOTION OF A PARTICLE IN AN EXTERNAL FIELD 189
CHAPTER IX. THE THEORY OF QUANTUM TRANSITIONS UNDER THE INFLUENCE OF AN EXTERNAL PERTURBATION 285
CHAPTER X. QUANTUM THEORY OF SYSTEMS CONSISTING OF IDENTICAL PARTICLES 336
CHAPTER XI. QUANTUM THEORY OF SCATTERING 374
CHAPTER XII. ELEMENTARY THEORY OF MOLECULES AND CHEMICAL BONDS 472
CHAPTER XIII. Basic IDEAS OF THE QUANTUM THEORY OF THE SOLID STATE 526
CHAPTER XIV. SECOND QUANTISATION OF SYSTEMS OF IDENTICAL BOSONS 553
CHAPTER XV. SECOND QUANTISATION OF SYSTEMS OF IDENTICAL FERMIONS 617

MATHEMATICAL APPENDICES
A. Some properties of the Dirac delta-function 654
B. The angular momentum operators in spherical coordinates 657
C. Linear operators in a vector space; matrices 658
D. Confluent hypergeometric functions; Bessel functions 664
E. Group theory 670

INDEX 675

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