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Stability And Oscillation Of Elastic Systems Modern Concepts, Paradoxes And Errors by Ya. G. Panovko; I. I. Gubanova

This book discusses contemporary problems such as “jumps” in elastic systems, problems of aeroelasticity, problems of frictional self-oscillations, and self-synchronization, providing only the elementary data on these topics.

The first part examines the stability of equilibrium shapes in elastic systems. It addresses stability loss in cases of similar equilibrium shapes, the disappearance of stable equilibrium forms, and the absence of any equilibrium states. The error made by Euler in analysing stability loss is highlighted, and Mises’ truss is used as an example of stability loss in cases of similar equilibrium shapes.

The second part focuses on problems related to oscillations of linear systems, including systems with a fractional number of degrees of freedom, as well as the free oscillations of a cantilever in the field of centrifugal forces. Four methods for solving the problem of the action of periodic instantaneous impulses are presented. The Tacoma catastrophe is analysed as an example of aeroelastic oscillations.

Finally, the book explores problems of nonlinear system oscillations, including the vibration maintenance of rotation, the Sommerfeld effect, and self-oscillations of a quasi-system with dry friction.

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Table of Contents

Foreword
Foreword to the First Edition …………………………………………….. v
Foreword to the Second Edition ……………………………………………. vi
Part 1: The Stability of Equilibrium Shapes of Elastic Systems
Introduction ………………………………………………………………….. 1

Chapter I: The Loss of Stability Upon the Appearance of Similar Equilibrium Shapes

§1. Euler’s Error ……………………………………………………………………. 5
§2. The Effect of Subcritical Compression of a Bar on the Critical Value of the Compressive Force …. 11
§3. One Version of the Application of the Energy Method ………………………. 17
§4. Loads Whose Values Depend on the Displacements ………………………….. 22
Chapter II: Loss of Stability Upon the Appearance of Non-Similar Equilibrium Shapes

§5. The Mises’ Truss ………………………………………………………………… 43
§6. The Stability of a Fluted Strip ………………………………………………….. 51
§7. More Examples of Systems with Jumps; Discussion of the Results …………. 58
Chapter III: Stability Loss Upon the Disappearance of Stable Equilibrium Shapes

§8. Tracking Loads: Static Statement of the Problem …………………………. 70
§9. Tracking Loads: Dynamic Statement of the Problem …………………….. 80
§10. Tracking Loads: A System with Two Degrees of Freedom ………………… 87
§11. The History of the Problem …………………………………………………… 91
Chapter IV: Stability Losses When Any Forms of Equilibrium Disappear

§12. General Stability of High Buildings ………………………………………… 97
§13. Characteristics of “Deformation Calculations” …………………………….. 104
§14. Two Discussions (Solutions of R. Lorenz and V. E. Vlasov) ……………….. 113
§15. Stability Losses of a Rod Under Tension …………………………………. 125
§16. Critical Internal Pressure for a Spherical Shell ……………………………. 136
§17. Rotation of a Flexible Shaft in a Rigid Tube-Shell ………………………… 140
Chapter V: Buckling of Not Fully Elastic Rods

§18. Elastic-Plastic Buckling: Classical Concept ……………………………….. 153
§19. Elastic-Plastic Buckling: Present-Day Concept ……………………………. 160
§20. Buckling of a Rod in a Statically Indeterminate System …………………. 167
§21. Stability Loss in the Case of Material Creep ……………………………….. 175
Part 2: Oscillations of Elastic Systems
Introduction …………………………………………………………………… 184

Chapter VI: Certain Problems of Oscillations of Linear Systems

§22. System with a Fractional Number of Degrees of Freedom ………………. 187

§23. Free Oscillations of a Cantilever in the Field of Centrifugal Forces ………. 192

§24. Equal-Frequency Shock Absorber ……………………………………………. 197

§25. Comments on the Formulas of Rayleigh and Grammel ……………………… 201

§26. Lagrange Errors ………………………………………………………………. 214

§27. Formula of A. N. Krylov ………………………………………………………. 224

§28. Four Methods of Solving the Problem of the Action of Periodic Instantaneous Impulses …….. 233

§29. Superpositions: Variations of Using It in Problems of Forced Oscillations … 244

§30. The “Inverse” Form of Differential Equations of Oscillations ……………. 252

§31. Terminology Information: Impedance, Receptance, Admittance, Response, Anti-Resonance …. 259

§32. Parametric Excitation of Oscillations ……………………………………… 269

§33. Destabilizing Action of the Forces of Viscous Friction …………………….. 285

§34. Linear Realisations of Dry Friction Forces …………………………………. 292

§35. Paradox Connected with Damping Coverings ……………………………… 302

§36. Damping of Pipeline Oscillations by Coriolis Forces ………………………. 309

Chapter VII: Dynamic Action of a Moving Load

§37. Brief Historical Sketch ……………………………………………………… 314
§38. Bresse Error ……………………………………………………………………… 322
§39. A Travelling Bending Wave …………………………………………………….. 327
§40. Action of an Infinite Strip of a Moving Load ………………………………. 332
Chapter VIII: Aeroelastic Oscillations

§41. Dynamic Problems of Aeroelasticity Theory ……………………………….. 338
§42. “Classical” Flutter ………………………………………………………………. 341
§43. Tacoma Catastrophe: Separation Flutter …………………………………… 351
Chapter IX: Problems of Nonlinear System Oscillations

§44. Vibration Maintenance of Rotation …………………………………………. 358
§45. Dynamics of the Boisse-Sarda Regulator …………………………………… 365
§46. Sommerfeld Effect ………………………………………………………………… 372
§47. Self-Oscillations: Method of Slowly Changing Amplitudes ……………….. 383
§48. Self-Oscillations of a Quasi-System with Dry Friction …………………… 393
§49. Discontinuous Self-Oscillations in the Case of Dry Friction ……………. 400
§50. Delta Method ……………………………………………………………………… 406

Theory Of Stellar Spectra by V.V. Sobolev

The only source of our information on stars is their radiation. From stellar spectra we judge the structure of stellar atmospheres, their chemical composition, and physical processes taking place there. Spectrograms are interpreted on the basis of the theory of stellar spectra, wherein lies its enormous value for astrophysics. Until recently astronomers could observe stellar radiation only in a very small range of frequencies making up the visible region of the spectrum. However, about 20 years ago radio astronomy came into being which permits investigating the radiation of celestial bodies in a completely different spectral region.

Quite recently, in connection with the launching of satellites and rockets, there arose the theoretical possibility of obtaining stellar spectra in any frequency range. So far, highly useful spectrograms of stars and the sun have been obtained in the so-called rocket ultraviolet. It is obvious that this broadening of observational data will even further increase the significance of the theory of stellar spectra. At the same time it is necessary to improve and extend this theory.

The surface layers of stars, out of which their spectra arise, represent
strongly ionized gases, i.e., plasma. Plasma studies are also being carried out in physics laboratories, having increased in intensity of late. Methods used by physicists in studying plasmas are in many respects similar to methods used by astrophysicists in studying stellar atmospheres. Therefore, the theory of stellar spectra is of interest not only to astrophysicists but also to physicists.

An excellent example of the broad interest in the theory of stellar spectra is the summer seminary on problems of this theory, organized by the
Astronomical Council of the Academy of Sciences of the USSR and the Leningrad University and held in Leningrad in June 1964* About 150 young astrophysicists and physicists of the Soviet Union participated in the sessions. This book was written on the basis of the lectures given at that time.
The first part of the book examines atomic processes associated with the
formation of spectra, with special emphasis on calculation of the energy levels of the atom and the probability of transitions between these levels. The second part deals with the theory of radiation transfer, which forms an important aspect of the theory of stellar spectra. The next two parts discuss the most essential problems in the formation of spectra of different types of stars and nebulae. The last part, devoted to ultraviolet spectra of celestial bodies, mainly gives a review of observational data and their qualitative interpretation (since, as yet, no quantitative theory of these spectra has been established).
The diversity of the problems of the modern theory of stellar spectra makes it impossible to present them with sufficient completeness in a single monograph.
The authors of this book have endeavored to acquaint the reader with the most important of these problems.

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Preface …………………………………………… iii
PART I. PHYSICAL PROCESSES CONNECTED WITH THE FORMATION OF SPECTRA ………… 1
Plasma Spectroscopy (S.E. Frish) …………………….. 1
Calculations of Atomic Energy Levels (A.P. Yutsis and Ya.I. Vizbarayte) …………… 23
Theory of Atomic Transitions (G.F. Drukarev) ……….. 35

PART II. THEORY OF RADIATIVE TRANSFER ……………………… 64
Certain Nonlinear Problems of the Theory of Radiative Transfer (V.A. Ambartsumyan) ……. 64
Radiative Diffusion in Gases (V.V. Sobolev) …………. 75
Determination of the Populations of Excited Levels in an Optically Thick Gas Layer (V.V. Ivanov) ……….. 92
Nonstationary Radiation Field (I.N. Minin) …………. 116
Randomized Problem of Diffuse Reflection (R.V. Ambartsumyan) ………… 135

PART III. SPECTRA OF FIXED STARS ……………………….. 140
Models of Stellar Atmospheres (V.V. Sobolev) ………. 140
Continuous Spectra of Hot White Dwarfs (A.K. Kolesov) ……. 147
Model Atmospheres of Main-Sequence Stars of Class M (V.G. Buslavskiy) …….. 152
Determination of the Chemical Composition of Stellar Atmospheres (A.A. Boyarchuk) ……….. 160

PART IV. SPECTRA OF NONSTATIONARY STARS AND INTERSTELLAR MATTER ………… 170
Spectra of Nonstationary (Variable) Stars (V.G. Gorbatskiy) ……….. 170
Analysis of the Emission Spectra of Nonstationary Stars (A.A. Boyarchuk) ………… 194
Spectra of Interstellar Matter (S.A. Kaplan) ………… 203
Radio Observations of Planetary Nebulae (Yu. N. Pariyskiy) ………. 216

PART V. SPECTRA OF CELESTIAL BODIES IN THE FAR ULTRAVIOLET REGION ……….. 220
Spectra of Celestial Bodies in the Far Ultraviolet Region (G.A. Gurzadyan) ………. 220

Psychological Research In The USSR Volume 1

A collection of articles on psychological research in the USSR.

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Contents

From the Editors ……………………………………………………………………. 7

PART ONE
General Problems of Psychology and Psychophysiology

Development of the Higher Mental Functions, by L. S. Vygotsky.
Translated by D. Myshne …………………………………………………………… 11

Problem of Development of the Higher Mental Functions ………….. 11
Method of Investigation ……………………………………………………….. 15
Structure of the Higher Mental Functions ………………………………… 21
Genesis of the Higher Mental Functions ………………………………….. 35

Problems of Psychological Theory, by S. L. Rubinstein.
Translated by D. Myshne …………………………………………………………… 59

Special Features of the Afferent Apparatus of the Conditioned Reflex and Their Importance to Psychology, by P. K. Anokhin.
Translated by D. Myshne …………………………………………………………… 67

Concerning the Decisive Role of Afferent Systems in Nervous Activity ………………………………………………………………………….. 68
Theory of Return Afferentation ………………………………………………. 71
Theory of the Acceptor of Action ……………………………………………. 81
Role of the Foregoing Theory in Explaining Certain Physiological and Mental Phenomena ……………………………………………………… 94

The Physiology of Conditioned Reflex, by E. A. Asratyan ………… 99

Bilateral Regulation as a Mechanism of Behaviour, by B. G. Ananyev.
Translated by D. Rottenberg …………………………………………………….. 128

Some Problems on the Control of Motor Acts, by N. A. Bernstein.
Translated by D. Danimanis ……………………………………………………… 139

Results of Experimental Studies on Properties of the Nervous System in Man, by B. M. Teplov and V. D. Nebylitsyn.
Translated by D. Rottenberg …………………………………………………….. 151

“Properties” and “Types” …………………………………………………….. 152
Syndromes of Basic Properties of the Nervous System …………….. 182
Correlation Between Mobility and Strength of the Nervous System …………………………………………………………………………… 186
Investigation of the “Inhibitory” Problem ……………………………….. 189
Electrical Activity of the Cerebral Cortex and Properties of the Nervous System ……………………………………………………………. 193
PART TWO
Sensations. Perceptions

On the Nature of Mental Reflection, by J. V. Shorokhova.
Translated by D. Myshne …………………………………………………………. 201

Interaction of the Sense Organs, by S. V. Kravkov.
Translated by D. Rottenberg …………………………………………………….. 225

Conclusion …………………………………………………………………………. 259

Manual Interaction in the Process of Tactile Perception, by B. F. Lomov.
Translated by D. Rottenberg …………………………………………………….. 260

General Characteristics of Perception …………………………………… 260
Manual Functional Asymmetries and Their Origin ………………….. 265
Experimental Data on Bimanual Interaction in Man ………………… 272
Specific Features of Bimanual Tactile Perception ……………………. 276
The Process of Bimanual Perception …………………………………….. 281
Digital Interaction in the Process of Bimanual Palpation …………. 287
Conditions Required for the Formation of an Integral Image in Bimanual Palpation ……………………………………………………….. 295
Synchronous Bimanual Palpation of Two Objects ………………….. 304

On the Basic Properties of the Mental Image and a General Approach to Their Analogue Simulation, by L. M. Wekker.
Translated by D. Myshne …………………………………………………………. 310

General Historical and Theoretical Premises in Stating the Problem of the Image ……………………………………………………… 310
Concerning Cybernetic Principles of the Theory of Image ………… 314
On the Control Function of the Image ……………………………………. 321
Concerning the Construction of a Mental Image …………………….. 326

Orienting Reflex as Information, by Y. N. Sokolov.
Translated by D. Myshne …………………………………………………………. 330

Criterion for Isolating the Orienting Reflex as an Independent Functional System ……………………………………………………….. 330
Orienting Reflex and Activation Reaction ……………………………… 335
Extinction of the Orienting Reflex and Associative Phenomena … 340
“Nerve Model of the Stimulus” ……………………………………………. 345
Algorithm of the Orienting Reaction During Differentiation of Sound Sequences ……………………………………………………….. 349
Algorithm of the Orienting Reaction in Tactile Perception ………. 359
Conclusion …………………………………………………………………………. 360

Concerning the Activity of Man’s Visual System, by A. N. Leontyev and Y. B. Gippenreiter.
Translated by D. Myshne …………………………………………………………. 361

Development of Perceptual Activity and Formation of a Sensory Image in the Child, by A. V. Zaporozhets and V. P. Zinchenko.
Translated by D. Myshne …………………………………………………………. 395

[Psychology] of Set, by R. Natadze.
Translated by D. Myshne

Method
Preliminary Experiments
The Basic Experiment
Results of the First Variant of the Basic Experiment
Results of the Second Variant of the Basic Experiment
Conclusions

Concerning Time Perception and the Feedback Principle, by D. G. Elkin.
Translated by D. Myshne

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

Measure And Derivative A Unified Approach by G.E. Shilov; B.L. Gurevich

This volume is intended as a textbook for students of
mathematics and physics, at the graduate or advanced
undergraduate level. It should also be intelligible to
readers with a good background in advanced calculus
and sufficient “mathematical maturity.”
The phrase “unified approach” in the title of the book
refers to the consistent use of the Daniell scheme, which
starts from the concept of an elementary integral defined
(axiomatically) on a family of elementary functions. In
the Introduction we explain in detail why we prefer
this approach to others, in particular to the Lebesgue-
Radon-Frechet approach, which starts from axiomatic
measure theory.

Revised English Edition
Translated and Edited by Richard A. Silverman

You can get the book here and here.

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