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Chemical Thermodynamics by M.Kh. Karapetyants

The book is primarily aimed at students in higher education specialising in chemistry, particularly future engineers. The author has avoided unnecessary abstraction and overly complex mathematics to ensure the material remains practical and accessible, while still providing a solid theoretical foundation. The content includes approximate laws that allow for quick, practical problem-solving, even when precise values are unavailable. The author integrates empirical thermodynamics with the periodic table to make thermodynamic concepts more comprehensible, particularly entropy, which students often find difficult to grasp.

The book also addresses the importance of connecting thermodynamics with other branches of chemistry, such as general and inorganic chemistry, to enhance students’ understanding for later courses. The primary focus is on the thermodynamics of gaseous systems, with less emphasis on solutions and electrolytes. Numerous examples, mainly related to inorganic substances and chemical processing, help students apply theory to practical problems, with calculations that can be compared to experimental data. The book also includes many tables and figures derived from various sources to support these applications.

Translated from the Russian by G. Leib

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Contents

List of Tables 11
Preface 13
Chapter 1. INTRODUCTION 15
1.1. The Subject and Method of Thermodynamics 15
1.2. Basic Concepts and Definitions 17
1.2.1. Systems and Their Classification 17
1.2.2. Thermodynamic Parameters 18
1.2.3. Work and Heat 22
1.2.4. Reversible and Irreversible Processes 23
1.2.5. Mathematical Relations Between the Parameters of State 28
1.3. Terms and Symbols 33
Chapter 2. THE FIRST LAW OF THERMODYNAMICS 35
2.1. Content of the First Law 35
2.1.1. Cyclic Processes 35
2.1.2. Non-Cyclic Processes. Internal Energy 36
2.2. Enthalpy 41
Chapter 3. HEAT EFFECTS AND HEAT CAPACITIES 45
3.1. Hess’s Law 45
3.2. Standard Heat Effects 49
3.3. Some Methods of Calculating Heat Effects 53
3.3.1. Heats of Formation 53
3.3.2. Heats of Combustion 57
3.3.3. Comparative Calculation of Heat Effects 58
3.4. Heat Capacity 58
3.4.1. Heat Capacity in Different Processes 58
3.4.2. Temperature Dependence of Heat Capacity 61
3.4.3. Certain Laws 71
3.5. Temperature Dependence of Heat Effect 73
3.5.1. Kirchhoff Equation 73
3.5.2. Equation AH = <p(T) in Its Final Form 77
3.5.3. Heat Balance 84
Chapter 4. THE SECOND LAW OF THERMODYNAMICS 87
4.1. Content of the Second Law 87
4.1.1. The Carnot Cycle 87
4.1.2. Thermodynamic Temperature Scale 91
4.1.3. Impossibility of a Perpetual Motion Machine 92
4.2. Entropy 94
4.2.1. Change in Entropy in Reversible Processes 95
4.2.2. Change in Entropy in Irreversible Processes 97
4.2.3. Change in Entropy as a Criterion of the Equilibrium and Spontaneity of Processes 98
4.2.4. Relation Between Entropy and Other Thermodynamic Parameters and Some Relationships Between Derived Functions 102
4.3. Substantiation of the Second Law 106
4.3.1. Thermodynamic Probability of a State 106
4.3.2. Phase Space 106
4.3.3. Relationship Between Entropy and Thermodynamic Probability 108
4.3.4. Fluctuations 110
4.3.5. The Invalidity of the “Theory of Heat Death” of the Universe 112
Chapter 5. THERMODYNAMIC AND CHEMICAL POTENTIALS. THE GENERAL CONDITIONS OF EQUILIBRIUM 114
5.1. Thermodynamic Potentials 114
5.1.1. Change in Thermodynamic Potential as a Criterion of the Equilibrium and Spontaneous Nature of a Process 119
5.1.2. Various Thermodynamic Relationships 121
5.2. Characteristic Functions 123
5.3. Chemical Potential 128
5.4. General Conditions of Equilibrium 131
5.4.1. Stable and Unstable Equilibria 132
5.4.2. Equilibrium Coexistence of Phases. The Gibbs Phase Rule 134
5.4.3. Principle of Displacement of Equilibrium 139
Chapter 6. ONE-COMPONENT HOMOGENEOUS SYSTEMS 141
6.1. Ideal Gas 141
6.2. Equations of State of a Real Gas 150
6.3. Fugacity 159
6.3.1. Standard State 160
6.3.2. Temperature Dependence of Fugacity 162
6.3.3. Methods of Calculating Fugacity 163
6.4. Throttling 168
6.5. Calculation of Properties of Gases According to Experimental Data 175
6.5.1. Calculations Using the Relationships p-V-T and Cp = q>(T) 175
6.5.2. Calculations Using the Relationships Cp = <p(p, T) or H = q(p, T) and VT> = 9 (p) 184
6.5.3. Calculations Using (ij and C9 185
6.5.4. Influence of Pressure on the Heat Effect of a Reaction 185
6.6. A Generalized Method of Calculating Selected Properties of Gases and Liquids at Pressures above Atmospheric 186
6.6.1. Gases 186
6.6.2. Liquids 199
Chapter 7. ONE-COMPONENT HETEROGENEOUS SYSTEMS 205
7.1. Relationship Between Temperature and Pressure with Coexisting Phases 205
7.1.1. Clapeyron-Clausius Equation 205
7.1.2. Approximate Relationships 209
7.2. Methods for the Comparative Calculation of the Temperature Dependence of the Saturated Vapour Pressure 214
7.2.1. Straight Line Method 215
7.2.2. Method of Comparing Boiling Points of Given and Standard Substances at Equal Vapour Pressures 216
7.2.3. Method of Comparing Vapour Pressures of Various Substances at Equal Boiling Points 218
7.2.4. Method of Comparing Vapour Pressures of Various Substances at Equal Reduced Boiling Points 220
7.3. Critical State 221
7.4. Heat Capacities of Coexisting Phases and Heats of Phase Transitions 227
7.4.1. Heat Capacities of Coexisting Phases 227
7.4.2. Heats of Phase Transitions 232
7.5. Influence of Total Pressure on Saturated Vapour Pressure 244
7.6. Influence of Surface Curvature on Saturated Vapour Pressure 247
7.7. Second-Order Phase Transitions 249
Chapter 8. SOLUTIONS 251
8.1. Fundamental Concepts and Definitions 251
8.2. Partial Molar Quantities 255
8.2.1. Basic Equations 257
8.2.2. Methods of Calculation 260
8.3. Heat Capacities and Enthalpies of Solutions 264
8.3.1. Partial Molar Heat Capacities 264
8.3.2. Partial Molar Enthalpies 265
8.4. Ideal Solutions 272
8.5. Infinitely Dilute Solutions 278
8.5.1. Partial Molar Quantities 279
8.5.2. Henry’s Law 281
Chapter 9. BINARY SOLUTION-PURE COMPONENT EQUILIBRIUM 285
9.1. Relationship Between Temperature and Concentration 285
9.1.1. Solution-Solid Component Equilibrium 287
9.1.2. Analysis of Solubility Diagrams 292
9.1.3. Solution-Gas Equilibrium 304
9.2. Relationship Between Pressure and Concentration 305
9.2.1. Solution-Solid Component Equilibrium 306
9.2.2. Solution-Gas Equilibrium 307
9.3. Gas Mixture-Pure Component Equilibrium 317
9.4. Influence of Dispersion on Solubility 318
Chapter 10. SOLUTION-SOLUTION EQUILIBRIUM IN BINARY MIXTURES 319
10.1. Liquid-Gas Equilibrium for Completely Miscible Liquids at Low Pressures 319
10.1.1. Ideal Solution-Mixture of Ideal Gases 319
10.1.2. Non-Ideal Solution-Mixture of Ideal Gases 322
10.1.3. Separation of Solution Components 332
10.2. Liquid-Gas Equilibrium for Completely Miscible Liquids at High Pressures 334
10.2.1. Critical Phenomena 341
10.3. Equilibrium in Systems with Incompletely Miscible Liquids 349
10.3.1. Liquid-Gas Equilibrium 349
10.3.2. Liquid-Liquid Equilibrium 351
10.3.3. Gas-Gas Equilibrium 352
10.4. Liquid-Gas Equilibrium for Immiscible Liquids 355
Chapter 11. EQUILIBRIUM IN THREE- AND FOUR-COMPONENT SYSTEMS 359
11.1. Depicting Composition 359
11.1.1. Three-Component Systems 359
11.1.2. Four-Component Systems 361
11.2. Liquid-Solid Equilibrium in Three-Component Systems 362
11.2.1. Substances Forming No Compounds 362
11.2.2. Substances Forming Compounds 366
11.2.3. Isotherms of Aqueous Solutions of Two Common-Ion Salts 367
11.3. Mutual Solubility of Three Liquids 385
11.4. Liquid-Gas Equilibrium in Ternary Systems 391
11.4.1. Isotherm 391
11.4.2. Is

obaric Systems 399
Chapter 12. THE PRINCIPLE OF MAXIMUM ENTROPY 405
12.1. Entropy as a Thermodynamic Function 405
12.2. Method of Maximum Entropy 410
12.3. Application of the Maximum Entropy Principle in Thermodynamics 414
Chapter 13. MODERN CONCEPTS OF THERMODYNAMICS 419
13.1. Thermodynamic Models 419
13.2. Relations with Other Areas of Science 421
13.3. Role of Thermodynamics in Physical Chemistry 424
13.4. Applications of Thermodynamics in Industry 427
13.5. Advanced Topics in Thermodynamics 430

Chapter 14. EQUILIBRIUM TRANSFORMATION 518

14.1. Direction of a Process 518

14.2. Calculation of Equilibrium Transformation 527

14.2.1. Reactions in the Gaseous Phase 528

14.2.2. Reactions in Solutions 531

14.2.3. Heterogeneous Reactions 533

14.2.4. Electrochemical Reactions 537

14.3. Influence of Various Factors on the Extent of a Reaction 541

14.3.1. Temperature 541

14.3.2. Pressure 545

14.3.3. Presence of an Inert Gas 548

14.3.4. Ratio of Reactants 549

14.3.5. Change in Surface Area 550

14.3.6. Kind of Reaction 552

14.4. Equilibrium in Complex Chemical Systems 553

14.5. Sources of Errors in Calculating Equilibrium 562

14.5.1. Errors Due to Inaccuracy of Experimental Data 562

14.5.2. Errors Connected with the Processing of Experimental Data 564

14.6. Theoretical and Practical Extents of a Reaction 566

Chapter 15. FUNDAMENTALS OF QUANTUM STATISTICAL CALCULATIONS OF THERMODYNAMIC FUNCTIONS AND CHEMICAL EQUILIBRIUM FROM SPECTROSCOPIC DATA 568

15.1. Introduction 568

15.2. Thermodynamic Properties of Gases Due to Translational Degrees of Freedom 572

15.3. Thermodynamic Properties of Gases Due to Intramolecular Degrees of Freedom 575

15.3.1. Rotational Partition Function 577

15.3.2. Vibrational Partition Function 583

15.3.3. Partition Function for Electronic Excitation 587

15.3.4. Nuclear Spin 588

15.3.5. Effect of Isotopic Composition 589

15.3.6. Group of Properties 589

15.4. Calculation of Chemical Equilibrium 592

APPENDICES 599

List of Symbols 599

Heat Capacities, Standard Enthalpies and Gibbs Energies of

Similarity And Dimensional Methods In Mechanics by L. I. Sedov

Dimensional analysis and similarity theory are essential in physics and engineering, particularly for designing and testing complex structures like airplanes, ships, and dams. These theories guide the conditions for model experiments and identify key parameters for fundamental effects and operations. Despite their simplicity and utility, they are often inadequately explained in textbooks and educational practices, leading to confusion and misconceptions.

The book highlights the importance of clear definitions of dimensional and dimensionless quantities and foundational concepts like the number of basic units of measurement. It critiques the superficial treatment of these topics in academia, which has occasionally led to paradoxes, such as misinterpretations in Rayleigh’s conclusions on heat emission.

Dimensional analysis is especially valuable when combined with broader physical principles, yielding significant insights in fields like turbulence, where a complete mathematical framework is lacking. The book includes new results in turbulence theory and provides detailed analyses of problems like turbulent fluid motion and Newton’s second law.

While many applications of dimensional analysis are not covered, the text aims to demonstrate standard methods and inspire the selection and formulation of new problems and experiments. The first half of the book is accessible to general readers, while the latter half requires some knowledge of hydromechanics.

Translated from the Russian by V. I. Kisin

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CONTENTS
Foreword to the First Russian Edition 7
Foreword to the Third Russian Edition 9
Foreword to the Sixth Russian Edition 10
Foreword to the Eighth Russian Edition 11
Foreword to the Ninth Russian Edition 11

CHAPTER I. General Dimensions Theory
§ 1. Introduction 13
§ 2. Dimensional and Dimensionless Quantities 14
§ 3. Fundamental and Derived Units of Measurement 15
§ 4. Dimensions Formulas 20
§ 5. On Newton’s Second Law 21
§ 6. Nature of the Functional Relations Between Physical Quantities 27
§ 7. Parameters Defining a Class of Phenomena 32
References 35

CHAPTER II. Similarity, Modelling, and Various Examples of the Application of Dimensional Analysis
§ 1. Motion of a Simple Pendulum 36
§ 2. Flow of a Heavy Fluid Through a Spillway 38
§ 3. Fluid Motion in Pipes 40
§ 4. Motion of a Body in a Fluid 44
§ 5. Heat Transfer from a Body in a Fluid Flow 51
§ 6. Dynamic Similarity and Modelling of Phenomena 54
§ 7. Steady Motion of a Solid Body in a Compressible Fluid 63
§ 8. Unsteady Motion in a Fluid 68
§ 9. Ship Motion 72
§ 10. Planing over the Water Surface 79
§ 11. Impact on Water 86
§ 12. Entry of a Cone and a Wedge at Constant Speed into a Fluid 93
§ 13. Small-Amplitude Waves on the Surface of an Incompressible Fluid 95
§ 14. Three-Dimensional Self-Similar Motions of Continuous Media 103
References 106

CHAPTER III. Applications to the Theory of Motion of a Viscous Fluid and to the Theory of Turbulence
§ 1. Diffusion of Vorticity in a Viscous Fluid 108
§ 2. Exact Solutions of the Equations of Motion of a Viscous Incompressible Fluid 110
§ 3. Boundary Layer in the Flow of a Viscous Fluid Past a Flat Plate 116
§ 4. Isotropic Turbulent Motion of an Incompressible Fluid 120
§ 5. Steady Turbulent Motion 151
References 163

CHAPTER IV. One-Dimensional Unsteady Motion of a Gas
§ 1. Self-Similar Motion of Spherical, Cylindrical, and Plane Waves in a Gas 166
§ 2. Ordinary Differential Equations and the Shock Conditions for Self-Similar Motions 175
§ 3. Algebraic Integrals for Self-Similar Motion 187
§ 4. Motions which Are Self-Similar in the Limit 196
§ 5. Investigation of the Family of Integral Curves in the (z, V) Plane 200
§ 6. The Piston Problem 208
§ 7. Problem of Implosion and Explosion at a Point 211
§ 8. Spherical Detonation 213
§ 9. Flame Propagation 220
§ 10. Collapse of an Arbitrary Discontinuity in a Combustible Mixture 225
§ 11. Problem of a Strong Explosion 229
§ 12. Point Explosion with Counterpressure Taken into Account 260
§ 13. On Modelling and on Formulas for the Peak Pressure and Impulse of Explosions 272
§ 14. Problem of a Strong Explosion in a Medium with a Variable Density 282
§ 15. Unsteady Motion of a Gas when the Velocity is Proportional to the Distance from the Centre of Symmetry 293
§ 16. On the General Theory of One-Dimensional Motion of a Gas 304
§ 17. Asymptotic Laws of Shock Wave Damping 317
References 325

CHAPTER V. Introduction to the Theory of Gas Engines
§ 1. On Averaging of Nonuniform Gas Flows in Ducts 334
§ 2. Similarity Conditions and Abstract Parameters Determining the Characteristics of Compressors 348
§ 3. On Flight Efficiency of an Ideal Propeller and an Ideal Air-Breathing Jet Engine 359
References 366

CHAPTER VI. Applications to Astrophysical Problems
§ 1. Some Observational Results 367
§ 2. On the Equations of Equilibrium and Motion of a Gaseous Mass Simulating a Star 377
§ 3. Theoretical Formulas Relating Luminosity with Mass, and Radius with Mass 382
§ 4. Some Simple Solutions of the System of Equations of Stellar Equilibrium 386
§ 5. On the Relation Between the Period of Variation of the Brightness and the Average Density for Cepheids 392
§ 6. On the Theory of the Flare-ups of Novae and Supernovae 395
References 417

Name Index 419
Subject Index 422

रासायनिक मूलद्रव्यांचा शोध – दमी. त्रीफोनोव, व. त्रीफोनोव (Chemical Elements How They Were Discovered In Marathi by D. N. Trifonov, V. D. Trifonov )

या पुस्तकामध्ये आपण रसायनशास्त्रातील “मुळाक्षरां “ची कशी रचना झाली, तसेच आपल्या चौकस बुद्धिमत्तेतून संशोधकांनी एकामागोमाग एकेका नव्या मूलद्रव्याचा शोध कसा लावला याचे विवेचन करणार आहोत.

बहुतेक सर्व रासायनिक मूलद्रव्यांवर अनेक पुस्तके लिहिली गेली आहेत. त्यांची संख्या एवढी प्रचंड आहे की त्यांचे एक स्वतंत्र ग्रंथालयच होईल. त्या पुस्तकांमध्ये, मूलद्रव्यांचा अंतर्भाव असणारे खडक व खनिजे, त्यांच्या नि- एकर्षणाच्या विविध पद्धती, मूलद्रव्यांचे भौतिक आणि रासायनिक गुणधर्म यांचा आढावा घेतलेला असतो. काही मूलद्रव्ये आश्चर्य वाटण्याएवढी मुबलक आहेत. त्यांचे उपयोगही अनपेक्षित वाटावेत एवढ्या भिन्नभिन्न क्षेत्रात केलेले आढळतील. खरे तर आजच्या प्रगत वैज्ञानिक युगात प्रत्येक मूलद्रव्याचा काहीना काहीतरी महत्त्वपूर्ण उपयोग आहेच आहे असेच दिसेल. प्रत्येक मूलद्रव्याचे त्याचे स्वतःचे असे खास “चरित्र” असून ती सारी आपापल्या परीने वैशिष्ट्यपूर्णही आहेत हे खरे की मूलद्रव्यांच्या शोधांच्या इतिहासाबाबत अद्यापीही बरीच संदिग्धता आहे व इतिहासकारांना न सुटलेले प्रश्न अजूनही सोडविता आलेले नाहीत. कुणी सांगावे, त्यांच्यापैकी एखादा- दुसरा इतिहासकार तुम्हीही असू शकाल !

अनुवाद : राजेंद्र सहस्रबुद्धे

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"How did #SovietAuthors treat the #Holocaust? How did it differ from work elsewhere? And what are some of the challenges translating these works into English?"

#HarrietMurav and #SashaSenderovich discuss the new collection of #shortStories they edited and translated, #InTheShadowOfTheHolocaust: Short Fiction by #JewishWriters from the #SovietUnion

https://www.euraknot.org/soviet-holocaust-literature/
#SovietJews #JewishHistory #WWII #SovietLiterature #HolocaustLiterature #books @bookstodon

விலங்கியல் – வ. ஷாலாயேவ், நி. ரீக்கவ் (Zoology In Tamil by V. Shalayev, N. Rykov)

A comprehensive textbook on zoology.

Translated from the Russian.

விலங்கியல் தொடர்பான முழுமையான பாடநூல்.ரஷிய மொழியில் இருந்து மொழிபெயர்க்கப்பட்டது.

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In The World Of Isotopes by V. Mezentsev

A little book describing basics of isotopes and their applications in various fields.

Translated from the Russian by George Yankovsky

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Contents

1. On the Threshold of a New Era 5
2. “Factories of Radioisotopes” 7
3. “Radioeyes” 12
4. Radioisotopes and Geological Prospecting 20
5. Tracer Atoms in Industry 22
6. Tracer Atoms in Chemistry 25
7. The Geological Clock 27
8. A Pocket Electric Station 32
9. The Atom in Agriculture 33
10. The Atom in Medicine 45

A Brief Course Of Higher Mathematics by V.A. Kudryavtsev

The aim of this text is to set forth the essentials of higher mathematics and their applications in various fields. At present higher mathematics serves as the theoretical foundation for most branches of the natural, applied and engineering sciences. Therefore, every natural scientist must necessarily master its methods to be able to apply them for practical purposes.

Translated from the Russian by Leonid Levant

Many thanks to Guptaji for the scans and Balram Sharmaji of Kamgaar Prakashan for making this book available.

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Contents

INTRODUCTION

Chapter 1. The Rectangular Coordinate System in the Plane and Its Application to Simple Problems
Sec. 1. Rectangular Coordinates of a Point in the Plane
Sec. 2. Transformation of Rectangular Coordinates
Sec. 3. The Distance Between Two Points in the Plane
Sec. 4. Dividing a Line Segment in a Given Ratio
Sec. 5. The Area of a Triangle
Exercises

Chapter 2. The Equation of a Line
Sec. 6. Sets
Sec. 7. The Method of Coordinates in the Plane
Sec. 8. The Line as a Set of Points
Sec. 9. The Equation of a Line in the Plane
Sec. 10. Constructing a Line on the Basis of Its Equation
Sec. 11. Some Elementary Problems
Sec. 12. Two Basic Problems of Plane Analytical Geometry
Sec. 13. Algebraic Lines
Exercises

Chapter 3. The Straight Line
Sec. 14. The Equation of a Straight Line
Sec. 15. The Angle Between Two Straight Lines
Sec. 16. The Equation of a Straight Line Passing Through a Given Point in a Given Direction
Sec. 17. The Equation of a Straight Line Passing Through Two Points (Two-Point Form)
Sec. 18. The Intercept Form of the Equation of a Straight Line
Sec. 19. The Point of Intersection of Two Straight Lines
Sec. 20. The Distance from a Point to a Straight Line
Exercises

Chapter 4. Second-Order Lines
Sec. 21. The Circle
Sec. 22. Central Second-Order Curves (Conics)
Sec. 23. Focal Properties of Central Curves of the Second Order
Sec. 24. The Ellipse as a Uniformly Compressed Circle
Sec. 25. The Asymptotes of a Hyperbola
Sec. 26. The Graph of Inverse Proportionality
Sec. 27. Noncentral Quadric Curves
Sec. 28. The Focal Property of the Parabola
Sec. 29. The Graph of a Quadratic Trinomial
Exercises

Chapter 5. Polar Coordinates. Parametric Equations of a Line
Sec. 30. Polar Coordinates
Sec. 31. Relationship Between Rectangular and Polar Coordinates
Sec. 32. Parametric Equations of a Line
Sec. 33. Parametric Equations of the Cycloid
Exercises

Chapter 6. Functions
Sec. 34. Constants and Variables
Sec. 35. The Concept of Function
Sec. 36. Simplest Functional Relations
1. Direct Proportional Relation
2. Linear Relation
3. Inverse Proportional Relation
4. Quadratic Relation
5. Sinusoidal Relation
Sec. 37. Methods of Representing Functions
1. The Analytical Method
2. The Tabular Method
3. The Graphical Method
Sec. 38. The Concept of Function of Several Variables
Sec. 39. Implicit Function
Sec. 40. Inverse Function
Sec. 41. Classification of Functions of One Argument
Sec. 42. The Graphs of the Basic Elementary Functions
Sec. 43. Interpolation of Functions
Exercises

Chapter 7. The Theory of Limits
Sec. 44. Real Numbers
Sec. 45. Errors of Approximate Numbers
Sec. 46. Limit of a Function
Sec. 47. One-Sided Limits of a Function
Sec. 48. Limit of a Sequence
Sec. 49. Infinitesimals
Sec. 50. Infinitely Large Quantities
Sec. 51. Basic Properties of Infinitesimals
Sec. 52. Basic Limit Theorems
Sec. 53. Some Tests for the Existence of the Limit of a Function
Sec. 54. The Limit of X
Sec. 55. The Number e
Sec. 56. Natural Logarithms
Sec. 57. Asymptotic Formulas
Exercises

Chapter 8. Continuity of Functions
Sec. 58. Increments of an Argument and a Function. Continuity of a Function
Sec. 59. Another Definition of the Continuity of a Function
Sec. 60. Continuity of Basic Elementary Functions
Sec. 61. Basic Theorems on Continuous Functions
Sec. 62. Evaluation of Indeterminacies
Sec. 63. Classification of the Points of Discontinuity of a Function
Exercises

Chapter 9. The Derivative of a Function
Sec. 64. A Tangent to a Curve – 159
Sec. 65. Velocity of a Moving Point – 161
Sec. 66. The Derivative Defined Generally – 163
Sec. 67. Other Applications of the Derivative – 166
Sec. 68. Relation Between the Continuity and Differentiability of a Function – 167
Sec. 69. The Notion of an Infinite Derivative – 169
Exercises – 169

Chapter 10. Basic Derivative Theorems
Sec. 70. Introductory Notes – 170
Sec. 71. The Derivatives of Certain Simple Functions – 170
Sec. 72. Basic Differentiation Rules – 174
Sec. 73. The Derivative of a Composite Function – 179
Sec. 74. The Derivative of an Inverse Function – 182
Sec. 75. The Derivative of an Implicit Function – 184
Sec. 76. The Derivative of a Logarithmic Function – 185
Sec. 77. A Logarithmic Derivative – 188
Sec. 78. The Derivative of an Exponential Function – 188
Sec. 79. The Derivative of a Power Function – 190
Sec. 80. The Derivatives of Inverse Trigonometric Functions – 191
Sec. 81. The Derivative of a Function Represented Parametrically – 193
Sec. 82. The Table of Differentiation Formulas – 194
Sec. 83. Derivatives of Higher Orders – 195
Sec. 84. Physical Meaning of the Second Derivative – 195
Exercises – 196

Chapter 11. Applications of Derivatives
Sec. 85. The Theorem About Finite Increments of a Function and Its Corollaries – 199
Sec. 86. Increase and Decrease of a Function of One Argument – 201
Sec. 87. L’Hospital’s Rule – 204
Sec. 88. Taylor’s Formula for a Polynomial – 208
Sec. 89. Binomial Formula – 210
Sec. 90. Taylor’s Formula for a Function – 211
Sec. 91. Maxima and Minima of a Function of One Variable – 213
Sec. 92. Concavity and Convexity of the Graph of a Function. Points of Inflection – 220
Sec. 93. Approximate Solution of Equations – 223
Sec. 94. Construction of Graphs of Functions – 227
Exercises – 230

Chapter 12. Differentials
Sec. 95. The Differential of a Function – 232
Sec. 96. Relation Between the Differential of a Function and Its Derivative. The Differential of the Independent Variable – 235
Sec. 97. The Geometrical Meaning of the Differential – 237
Sec. 98. The Physical Meaning of the Differential – 237
Sec. 99. Approximate Calculation of Small Increments of a Function – 238
Sec. 100. Equivalence of the Increment and Differential of a Function – 239
Sec. 101. Properties of the Differential – 242
Sec. 102. Differentials of Higher Orders – 245
Exercises – 247

Chapter 13. Indefinite Integral
Sec. 103. Antiderivative. Indefinite Integral – 248
Sec. 104. Basic Properties of the Indefinite Integral – 251
Sec. 105. Table of Simplest Indefinite Integrals – 253
Sec. 106. Independence of the Form of an Indefinite Integral of the Argument Chosen – 254
Sec. 107. Basic Integration Methods – 258
Sec. 108. Techniques for Integrating Rational Fractions with a Quadratic Denominator – 263
Sec. 109. Integration of Simplest Irrational Expressions – 267
Sec. 110. Integration of Trigonometric Functions – 269
Sec. 111. Integration of Certain Transcendental Functions – 271
Sec. 112. Cauchy’s Theorem. Some Important Integrals Inexpressible in Terms of Elementary Functions – 271
Exercises – 272

Chapter 14. The Definite Integral
Sec. 113. The Concept of the Definite Integral – 275
Sec. 114. A Definite Integral with a Variable Upper Limit – 277
Sec. 115. Geometrical Meaning of the Definite Integral – 279
Sec. 116. Physical Meaning of the Definite Integral – 281
Sec. 117. Basic Properties of the Definite Integral – 282
Sec. 118. The Mean Value Theorem – 286
Sec. 119. Integration by Parts in the Definite Integral – 288
Sec. 120. Change of Variable in the Definite Integral (Integration by Substitution) – 289
Sec. 121. The Definite Integral as the Limit of an Integral Sum – 291
Sec. 122. Approximate Evaluation of Definite Integrals – 293
Sec. 123. Simpson’s Formula – 296
Sec. 124. Improper Integrals – 297
Exercises – 299

Chapter 15. Applications of the Definite Integral
Sec. 125. Areas in Rectangular Coordinates – 301
Sec. 126. Areas in Polar Coordinates – 305
Sec. 127. The Arc Length in Rectangular Coordinates – 307
Sec. 128. The Arc Length in Polar Coordinates – 313
Sec. 129. Computing the Volume of a Solid by Known Cross Sections – 314
Sec. 130. The Volume of a Solid of Revolution – 316
Sec. 131. The Work of a Variable Force – 319
Sec. 132. Other Applications of the Definite Integral in Physics – 320
Exercises – 322

Chapter 16. Complex Numbers
Sec. 133. Arithmetic Operations on Complex Numbers – 325
Sec. 134. The Complex Plane – 326
Sec. 135. Theorems on the Modulus and Argument – 328
Sec. 136. Taking the Root from a Complex Number – 329
Sec. 137. The Concept of a Function of a Complex Variable – 331
Exercises – 332

Chapter 17. Determinants of Second and Third Order
Sec. 138. Second-Order Determinants – 335
Sec. 139. A System of Two Homogeneous Equations in Three Unknowns – 335
Sec. 140. Third-Order Determinants – 337
Sec. 141. Basic Properties of Determinants – 339
Sec. 142. A System of Three Linear Equations – 342
Sec. 143. A Homogeneous System of Three Linear Equations – 344
Sec. 144. A System of Linear Equations in Many Unknowns. Gauss’ Method – 346
Exercises – 349

Chapter 18. Fundamentals of Vector Algebra
Sec. 145. Scalars and Vectors – 351
Sec. 146. The Sum of Several Vectors – 352
Sec. 147. The Difference of Vectors – 353
Sec. 148. Multiplication of a Vector by a Scalar – 353
Sec. 149. Collinear Vectors – 354
Sec. 150. Coplanar Vectors – 355
Sec. 151. The Projection of a Vector on an Axis – 356
Sec. 152. The Rectangular Cartesian Coordinates in Space – 359
Sec. 153. The Length and Direction of a Vector – 360
Sec. 154. The Distance Between Two Points in Space – 361
Sec. 155. Operations on Vectors Represented in the Coordinate Form – 362
Sec. 156. Scalar Product of Two Vectors – 364
Sec. 157. Scalar Product of Vectors in the Coordinate Form – 366
Sec. 158. Vector Product of Vectors – 367
Sec. 159. Vector Product in the Coordinate Form – 369
Sec. 160. Triple Scalar Product – 371
Exercises – 373

Chapter 19. Fundamentals of Solid Analytic Geometry
Sec. 161. The Equations of a Surface and a Line in Space – 374
Sec. 162. The General Equation of a Plane – 380
Sec. 163. Angle Between Two Planes – 382
Sec. 164. Equations of a Straight Line in Space – 383
Sec. 165. The Derivative of a Vector Function – 387
Sec. 166. The Equation of a Sphere – 389
Sec. 167. The Equation of an Ellipsoid – 391
Sec. 168. The Equation of a Paraboloid of Revolution – 392
Exercises – 393

Chapter 20. Functions of Several Variables
Sec. 169. The Concept of a Function of Several Variables – 395
Sec. 170. Continuity – 398
Sec. 171. Partial Derivatives of the First Order – 401
Sec. 172. The Total Differential of a Function – 403
Sec. 173. Application of the Differential of a Function to Approximate Computations – 409
Sec. 174. Directional Derivatives – 410
Sec. 175. The Gradient – 413
Sec. 176. Partial Derivatives of Higher Orders – 417
Sec. 177. Test for the Total Differential – 418
Sec. 178. The Extremum (Maximum or Minimum) of a Function of Several Variables – 420
Sec. 179. An Absolute Extremum of a Function – 422
Sec. 180. Constructing Empirical Formulas by the Method of Least Squares – 424
Exercises – 428

Chapter 21. Series
Sec. 181. Examples of Infinite Series – 430
Sec. 182. Convergence of a Series – 431
Sec. 183. A Necessary Condition for Convergence of a Series – 435
Sec. 184. Comparison Tests – 437
Sec. 185. D’Alembert’s Test for Convergence – 440
Sec. 186. Absolute Convergence – 444
Sec. 187. Alternating Series. Leibniz’ Test – 446
Sec. 188. Power Series – 447
Sec. 189. Differentiation and Integration of Power Series – 450
Sec. 190. Expanding a Given Function into a Power Series – 450
Sec. 191. Maclaurin’s Series – 452
Sec. 192. Applying Maclaurin’s Series to Expanding Some Functions into Power Series – 453
Sec. 193. Applying Power Series to Approximate Calculations – 456
Sec. 194. Taylor’s Series – 459
Sec. 195. Series in a Complex Domain – 462
Sec. 196. Euler’s Formulas – 463
Sec. 197. Fourier Trigonometric Series – 464
Sec. 198. The Fourier Series of Even and Odd Functions – 473
Sec. 199. The Fourier Series of Nonperiodic Functions – 475
Exercises – 479

Chapter 22. Differential Equations
Sec. 200. Basic Concepts – 481
Sec. 201. Differential Equations of the First Order – 484
Sec. 202. First-Order Equations with Variables Separable – 486
Sec. 203. Homogeneous Differential Equations of the First Order – 492
Sec. 204. Linear Differential Equations of the First Order – 495
Sec. 205. Euler’s Method – 500
Sec. 206. Differential Equations of the Second Order – 502
Sec. 207. Integrable Types of Second-Order Differential Equations – 504
Sec. 208. Reducing the Order of a Differential Equation – 510
Sec. 209. Integrating Differential Equations with the Aid of Power Series – 513
Sec. 210. Common Properties of the Solutions of Second-Order Linear Homogeneous Differential Equations – 514
Sec. 211. Second-Order Linear Homogeneous Differential Equations with Constant Coefficients – 517
Sec. 212. Second-Order Linear Nonhomogeneous Differential Equations with Constant Coefficients – 523
Sec. 213. Differential Equations Containing Partial Derivatives – 533
Sec. 214. Linear Differential Equations with Partial Derivatives – 536
Sec. 215. Deriving the Heat Conduction Equation – 538
Sec. 216. The Problem on Temperature Distribution in a Limited Rod – 540
Exercises – 543
Chapter 23. Line Integrals
Sec. 217. The Line Integral of the First Kind – 546
Sec. 218. The Line Integral of the Second Kind – 548
Sec. 219. The Physical Meaning of the Line Integral of the Second Kind – 552
Sec. 220. Condition Under Which the Line Integral of the Second Kind is Independent of Path – 554
Sec. 221. The Work Performed by a Potential Force – 556
Exercises – 557

Chapter 24. Double and Triple Integrals
Sec. 222. Double Integrals – 561
Sec. 223. The Double Integral in Rectangular Cartesian Coordinates – 564
Sec. 224. Expressing a Double Integral in Polar Coordinates – 571
Sec. 225. The Euler-Poisson Integral – 575
Sec. 226. Mean-Value Theorem – 576
Sec. 227. Geometrical Applications of the Double Integral – 578
Sec. 228. Physical Applications of the Double Integral – 579
Sec. 229. Triple Integrals – 584
Exercises – 588

Chapter 25. Fundamentals of the Theory of Probability
A. Basic Definitions and Theorems
Sec. 230. Random Events – 591
Sec. 231. Algebra of Events – 593
Sec. 232. The Classical Definition of Probability – 594
Sec. 233. The Statistical Definition of Probability – 597
Sec. 234. The Theorem on Addition of Probabilities – 598
Sec. 235. A Complete Group of Events – 599
Sec. 236. The Theorem on Multiplication of Probabilities – 600
Sec. 237. Bayes’ Formula – 603

B. Repeated Independent Trials
Sec. 238. Elements of Combinatorial Analysis – 604
Sec. 239. The Formula of Total Probability – 605
Sec. 240. The Binomial Law of Distribution of Probabilities – 607
Sec. 241. The Laplace Local Theorem – 608
Sec. 242. The Laplace Integral Theorem – 610
Sec. 243. Poisson’s Theorem – 614

C. Random Variables and Their Numerical Characteristics
Sec. 244. A Random Discrete Variable and Its Distribution Law – 615
Sec. 245. Mathematical Expectation – 617
Sec. 246. Basic Properties of Mathematical Expectation – 618
Sec. 247. Variance – 621
Sec. 248. Continuous Random Variables. Distribution Functions – 626
Sec. 249. Numerical Characteristics of a Continuous Random Variable – 630
Sec. 250. Uniform Distribution – 631
Sec. 251. Normal Distribution – 633
Exercises – 636

Chapter 26. The Concept of Linear Programming
Sec. 252. An n-Dimensional Vector Space – 639
Sec. 253. Sets in n-Dimensional Space – 641
Sec. 254. The Problem of Linear Programming – 645

APPENDICES
A. Most Important Constants – 650
B. List of Formulas (Classified and Explained) – 650
I. Plane Analytic Geometry – 650
II. Differential Calculus—Functions of One Variable – 652
III. Integral Calculus – 654
IV. Complex Numbers, Determinants, and Systems of Simultaneous Equations – 658
V. Elements of Vector Algebra – 660
VI. Solid Analytic Geometry – 661
VII. Differential Calculus—Functions of Several Variables – 662
VIII. Series – 663
IX. Differential Equations – 666
X. Line Integrals – 668
XI. Double and Triple Integrals – 669
XII. Probability Theory – 671

ANSWERS – 674
SUBJECT INDEX – 684

Problems In Mathematics With Hints And Solutions by V. Govorov; P. Dybov; N. Miroshin; S. Smirnova

The book contains more than three thousand mathematics problems and covers each topic taught at school. The problems were contributed by 120 of the higher schools of the USSR and all the universities.

The book is divided into, four parts: algebra and trigonometry, fundamentals of analysis, geometry and vector algebra, and the problems and questions set during oral examinations. The authors considered it necessary to include some material relating to complex numbers, combinatorics, the binomial theorem, elementary trigonometric inequalities, and set theory and the method of coordinates. The authors believe that this material will help the readers systematize their knowledge in the principal divisions of mathematics.

In writing the book, the authors have used their experience of examining students in mathematics at higher schools and the preparation of television courses designed to help students revise their knowledge for the entrance examinations to higher educational establishments.

To make it easier for readers to grasp the material, some of the sections have been supplemented with explanatory text. The problems are all answered and some have additional hints or complete solutions.

The more difficult problems are marked with asterisks. Part 4 is entitled “Oral Examination Problems and Questions” and includes samples suggested by the higher schools.

The authors hope that this book will help those who want to enter the various types of higher school, aid the teachers, and be of use to all those who want to deepen and systematize their knowledge of mathematics.

EDITED BY PROF. A.I. PRILEPKO, D.Sc.

Translated from the Russian by Irene Aleksanova

You can get the book here and here.

Twitter: @MirTitles
Mastodon: @[email protected]
Mastodon: @[email protected]
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Fork us at: https://gitlab.com/mirtitles

Contents

Preface 5
Part 1 Algebra Trigonometry and Elementary Functions 9
1.1 Problems on Integers Criteria for Divisibility 9
1.2 Real Numbers Transformation of Algebraic Expressions 13
1.3 Mathematical Induction Elements of Combinatorics Binomial Theorem
1.4 Equations and Inequalities of the First and the Second Degree
1.5 Equations of Higher Degrees Rational Inequalities
1.6 Irrational Equations and Inequalities
1.7 Systems of Equations and Inequalities
1.8 The Domain of Definition and the Range of a Function
1.9 Exponential and Logarithmic Equations and Inequalities
1.10 Transformations of Trigonometric Expressions Inverse Trigonometric Functions
1.11 Solution of Trigonometric Equations Inequalities and Systems of Equations
1.12 Progressions
1.13 Solution of Problems on Derivation of Equations
1.14 Complex Numbers
Part 2 Fundamentals of Mathematical Analysis
2.1 Sequences and Their Limits An Infinitely Decreasing Geometric Progression Limits of Functions
2.2 The Derivative Investigating the Behaviour of Functions with the Aid of the Derivative
2.3 Graphs of Functions
2.4 The Antiderivative The Integral The Area of a Curvilinear Trapezoid
Part 3 Geometry and Vector Algebra
3.1 Vector Algebra
3.2 Plane Geometry Problems on Proof
3.3 Plane Geometry Construction Problems
3.4 Plane Geometry Calculation Problems
3.5 Solid Geometry Problems on Proof
3.6 Solid Geometry Calculation Problems
Part 4 Oral Examination Problems and Questions 241
4.1 Sample Examination Papers 241
4.2 Problems Set at an Oral Examination 244
Hints and Answers 265
Appendix 386

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A Collection Of Problems On A Course Of Mathematical Analysis by G. N. Berman

THE present Collection of Problems is intended for students studying mathematical analysis within the framework of a technical college course. In the arrangement of the material, the style of the exposition and basic pedagogical tendencies the Collection is most suited to the widely used Course of Mathematical Analysis of A. F. Bermant. At the same time, since the book contains systematically selected problems and exercises on the main branches of a Technical College course of mathematical analysis, it forms a useful adjunct independently of the text-book on which the course is based.

Translated by D. E. Brown
Translation edited by Ian N. Sneddon

You can get the book here and here.

Twitter: @MirTitles
Mastodon: @[email protected]
Mastodon: @[email protected]
Bluesky: mirtitles.bsky.social

#analysis #mathematics #problemBooks #sovietLiterature