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Higher Mathematics For Beginners And Its Application To Physics – Zeldovich
In this post, we will see the book Higher Mathematics For Beginners And Its Application To Physics by Ya. B. Zeldovich.
About the book
The title of this book gives the clue to our main aim, which is to initiate the reader into the realm of differential and integral calculus and, by applying these methods to the more important divisions of physics, to demonstrate the significance and power of higher mathematics.In this book, the student is regarded as a friend and ally who puts his faith in the teacher and the textbook and wishes ardently to make use of and apply to nature and technology the mathematical techniques offered to him. Comprehension of the subject expands as the result of analyzing examples and applications. In the strictly logical approach, the question of the significance and usefulness of the theorems studied remains in the background. In the present text, by contrast, we bring to the fore the mathematical ideas and their relationship with the study of nature.The notorious pitting of poets against physicists (mathematicians too) is a figment of the imagination of the poet B. Slutsky. In mathematics there is more poetry than any poet ever imagined. The history of science is proof that good mathematics is prophetic: mathematical analysis of the known opens up the path into the realm of the unknown and leads to new physical notions.In “Higher Mathematics for Beginners” I strove towards a constructive approach, to the eliciting of the meaning and aims of mathematical concepts and attempted, at least in part, to convey the spirit of the heroic period when these notions were born.The last two chapters (Dirac’s Remarkable Delta Function and What Next) are entirely different from the remainder of the book. The style too is quite changed. The aim there is to give the reader a feeling (of necessity, very superficial) of what complicated things lie ahead.
Translated from the Russian by George Yankovsky. First published 1973, revised from the 1972 Russian edition by Mir Publishers..
PS: This is a previous version of the book by Zeldovich and Msykis we had seen earlier.
You can get the book here.
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Contents
PREFACE TO THE FIFTH RUSSIAN EDITION 9
CHAPTER 1 FUNCTIONS AND GRAPHS 13
The functional relationship
Coordinates
Geometric quantities expressed in terms of coordinates
Graphical representation of functions. The equation of the straight line
The parabola
The cubic parabola, hyperbola, and circle
Altering the scale of a curve
Parametric representation of a curve
CHAPTER 2 THE CONCEPTS OF A DERIVATIVE AND AN INTEGRAL 45
Motion, distance and velocity
The derivative of a function as the limit of a ratio of increments
Notation of derivatives. The derivative of a power function
Approximating the values of a function by means of a derivative
A tangent to a curve
Increase and decrease of functions. Maximum and minimum
The area under a curve and determining distance from the rate of motion
The definite integral
The relationship between the integral and the derivative (Newton-Leibniz theorem)
The integral of a derivative
The indefinite integral
Properties of integrals
Mean values
Examples of derivatives and integrals
Summary
CHAPTER 3 COMPUTATION OF DERIVATIVES AND INTEGRALS 106
The differential sign. The derivative of a sum of functions
The derivative of an inverse function
The composite function
The derivative of a product of functions
The power function
The derivatives of algebraic functions with constant exponents
The exponential function
The number e
Logarithms
Trigonometric functions
Inverse trigonometric functions
The derivative of an implicit function
Integrals. Statement of the problem
Elementary integrals
General properties of integrals
Change of the variable in a definite integral
Series
Computing the values of functions by means of series
Condition for applicability .of series. The geometric progression
The binomial theorem for integral and fractional exponents
The order of increase and decrease of functions
CHAPTER 4 THE APPLICATION OF DIFFERENTIAL AND INTEGRAL CALCULUS TO GEOMETRY AND THE INVESTIGATION OF FUNCTIONS 174
Investigating maxima and minima of functions with the aid of the second derivative
Other types of maxima and minima. Salient points and discontinuities
Computing areas
Mean values
Arc length and curvature
Approximation of are length
Computing volumes, The volume and surface area of a solid of revolution
Curve sketching
CHAPTER 5 WATER FLOW. RADIOACTIVE DECAY AND NUCLEAR FISSION. ABSORPTION OF LIGHT 211
Water flow from a vessel, Statement of the problem
The solution of an equation when the derivative depends on the desired function
Radioactive decay
Measuring the mean lifetime of radioactive atoms
Series disintegration (radioactive family)
Investigating the solution for a radioactive family (series)
The chain reaction in the fission of uranium
Multiplication of neutrons in a large system
Escape of neutrons
Critical mass
Subcritical and supercritical mass for a constant source of neutrons
The critical mass
Absorption of light. Statement of the problem and a rough estimate
The absorption equation and its solution
Relationship between exact and approximate calculations
Effective cross-section
Attenuation of a charged-particle flux of alpha and beta rays
CHAPTER 6 MECHANICS 258
Force, work and power
Energy
Equilibrium and stability
Newton’s second law
Impulse
Kinetic energy
Motion under the action of a force dependent solely on the velocity
Motion under the action of an elastic force
Oscillations
Oscillation energy. Damped oscillations
Forced oscillations and resonance
On exact and approximate solutions of physical problems
Jet propulsion and Tsiolkovsky’s formula
The path of a projectile
The mass, centre of gravity and moment of inertia of a rod
The oscillations of a suspended rod
CHAPTER 7 THE THERMAL MOTION OF MOLECULES AND THE DISTRIBUTION OF AIR DENSITY IN THE ATMOSPHERE 344
The condition for equilibrium in the atmosphere
The relationship between density and pressure
Density distribution
The molecular kinetic theory of density distribution
The Brownian movement and kinetic-energy distribution of molecules
Rates of chemical reactions
Evaporation. The emission current of a cathode
CHAPTER 8 ELECTRIC CIRCUITS AND OSCILLATORY PHENOMENA IN THEM 361
Basic concepts and units of measurement
Discharge of a capacitor through a resistor
Oscillations in a capacitance circuit with spark gap
The energy of a capacitor
Inductance circuit
Breaking an inductance circuit
The energy of inductance
The oscillatory circuit
Damped oscillations
The_case of a large resistance
Alternating current
Mean quantities, power and phase shift
An alternating-current oscillatory circuit. Series resonance
Inductance and capacitance in parallel. Parallel resonance
Displacement current and the electromagnetic theory of light
Nonlinear resistance and the tunnel diode
CHAPTER 9 DIRAC’S REMARKABLE DELTA FUNCTION 422
Various ways of defining a function
Dirac and his function
Discontinuous functions and their derivatives
Representing the delta function by formulas
Application of the delta function
CONCLUSION. WHAT NEXT? 440
ANSWERS AND SOLUTIONS 445
APPENDIX 474
INDEX 481
Puntos Fijos (Lecciones Populares De Matemáticas) – Shashkin (Español)
En este post, veremos el libro Puntos Fijos (Lecciones Populares De Matemáticas)- por Yu. Shashkin. (Fixed Points)
Sobre el libro
La teoría de los puntos fijos encuentra sus raíces en el trabajo de Poincaré, Brouwer y Sperner y hace un uso extensivo de nociones topológicas como la continuidad, la compacidad, la homotopía y el grado de mapeo. Los teoremas de punto fijo tienen numerosas aplicaciones en matemáticas; la mayoría de los teoremas que aseguran la existencia de soluciones para ecuaciones diferenciales, integrales, de operador u otras ecuaciones se pueden reducir a teoremas de punto fijo. Además, estos teoremas se utilizan en áreas tales como la economía matemática y la teoría de juegos. Este libro presenta una exposición legible de la teoría de puntos fijos. El autor se centra en el problema de si un intervalo cerrado, cuadrado, disco o esfera tiene la propiedad de punto fijo.Otro objetivo del libro es mostrar cómo la teoría de puntos fijos utiliza ideas combinatorias relacionadas con la descomposición (triangulación) de figuras en partes distintas llamadas caras (símplexes), que se unen entre sí de manera regular. Se explican todos los conceptos básicos necesarios, como la continuidad, la compacidad, el grado de un mapa, etc., lo que hace que el libro sea accesible incluso para los estudiantes de secundaria. Además, el libro contiene ejercicios y descripciones de aplicaciones. Los lectores apreciarán este libro por su presentación lúcida de este tema matemático fundamental.
El libro fue traducido del ruso por ingeniero Antonio Molina García.
El libro fue publicado por la editorial Mir en 1991.
Créditos al cargador original.
Puedes conseguir el libro aquí.
(Disculpas por cualquier error, estoy usando la traducción automática)
Posted in books, Editorial Mir, mathematics, popular lectures in mathematics, soviet
Tagged combinatorio, conbinatorio, Editorial Mir, Español, Homeomorfismos., Homotopia, Lecciones Populares De Matemáticas, Lema de Sperner, Retracción, teorema de Brouwer, Transformaciones continuas
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Característica Euleriana (Lecciones Populares De Matemáticas) – Yu. Shashkin (Español)
En este post, veremos el libro Característica Euleriana (Lecciones Populares De Matemáticas) por Yu. Shashkin. (Euler Characteristic)
Sobre el libro
Este folleto da pruebas de la famosa fórmula de Euler para politopos convexos y de sus análogos para otras figuras (planos, espacios y polígonos). Las fórmulas llevan al lector de forma natural a la noción de característica de Euler. Se dan dos definiciones de la noción y se demuestra su equivalencia. Se discute el papel desempeñado por la característica de Euler en diferentes problemas geométricos, es decir, en la descomposición de planos y espacios, en el cálculo de áreas, en el recubrimiento de esferas.El libro está dirigido a alumnos de último año, estudiantes universitarios y universitarios y a todos los amantes de las matemáticas.
El libro fue traducido del ruso por B. Mirchevski.
El libro fue publicado por la editorial Mir en 1989.
Créditos al cargador original.
Puedes conseguir el libro aquí.
(Disculpas por cualquier error, estoy usando la traducción automática)
L’algèbre Récréative (Science Pour Tous) – Yakov Pérelman (Perelman)
Dans cet article, nous verrons le livre L’algèbre Récréative par Yakov Pérelman.
À propos du livre
II ne faut pas considérer ce livre comme un manuel d’ « algèbre facile >> pour débutants. Comme mes autres ouvrages de la même collection, L’algèbre récréative n’est pas un livre d’études, mais un livre à lire pour se distraire. Le lecteur auquel il est destiné doit avoir certaines connaissances d’algèbre, même si elles sont vagues ou à moitié oubliées. L’algèbre récréative a pour but de rendre plus précises, de rafraîchir et d’approfondir ces connaissances éparses et peu solides, mais surtout de former chez le lecteur le goût pour l’étude de l’algèbre et le désir de combler lui même, à l’aide des livres appropriés, ses lacunes éventuelles. Sous ce rapport, L’algèbre récréative n’a pas le même but qu’un livre comme Les nombres et les figures de Rademacher et Toeplitz, par exemple, qui ne demande pas au lecteur de «se rappeler les mathématiques que nous avons étudiées dans notre jeunesse». Mon livre, au contraire, cherche à consolider les connaissances acquises à l’école.Pour donner plus d’attrait et d’intérêt au sujet, j’ai utilisé différents moyens ; on trouvera ici des problèmes peu courants ou curieux, des incursions dans l’histoire des mathématiques, des applications inattendues de l’algèbre dans la vie pratique, etc.Par les matières étudiées, ce livre ne dépasse pas le programme d’algèbre de l’enseignement secondaire, abordant presque toutes les parties de celui ci, mais évitant les questions théoriques difficiles.
Le livre a été traduit du russe par Ch. Bir.
L’édition française a été publiée en 1984 par les éditions Mir.
Crédits à l’uploader d’origine.
Vous pouvez obtenir le livre ici et ici.
(J’ai utilisé la traduction automatique, toutes mes excuses pour les erreurs.)
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Ajoutez de nouvelles entrées au catalogue de livres détaillé ici.
Contenu
Extrait de la préface de l’auteur à Ia troisième édition 5
Chapitre premier. La cinquième opération mathématique 7
Qu’est-ce que la cinquième opération? 7
Chiffres astronomiques 8
Combien pèse l’air entier ? 10
Combustion « sans flamme et sans chaleur » 11
La diversité du temps 13
La serrure à secret 14
Un cycliste superstitieux 16
Résultats de doublements répétés 17
100 mille fois plus vite 19
10 mille opérations par seconde 23
Le nombre de parties d’échecs possibles 27
Le secret de l’automate joueur d’échecs 28
A l’aide de trois 2 32
A l’aide de trois 3 33
A l’aide de trois 4 33
A l’aide de trois chiffres identiques 34
A l’aide de quatre unités 35
A l’aide de quatre 2 36
Chapitre II. Le langage de l’algèbre 38
L’art de poser des équations 38
La vie de Diophante 40
Le cheval et le mulet 41
Les quatre frères 43
Des oiseaux sur les rives du fleuve 44
Une promenade 46
L’équipe de faucheurs 47
Les vaches dans le pré 51
Le problème de Newton 54
Permutation des aiguilles d’une montre 56
Coïncidence des aiguilles d’une montre 60
Les sept joueurs 61
rs absurdité apparente 63
ie pense pour nous 64
Quelques cas curieux 64
Dans un salon de coiffure 68
Le tramway et le piéton 69
Le vapeur et le radeau 70
Les deux boîtes de café 72
Une soirée dansante 73
Une reconnaissance navale 74
Au vélodrome 76
Une compétition de motocyclistes 77
La vitesse moyenne d’une automobile 79
Machines mathématiques 81
Chapitre III. Au secours de l’arithmétique
Multiplication instantanée 94
Les chiffres 1, 5 et 6 97
Les nombres 25 et 76 98
« Nombres » infinis 98
Le supplément 102
Divisibilité par 11 104
Le numéro de la voiture 106
Divisibilité par 108
Le théorème de Sophie Germain 109
Nombres divisibles 110
Nombre des nombres premiers 112
Le plus grand nombre premier connu 113
Un calcul sérieux 113
L’algèbre ne simplifie pas toujours 117
Chapitre IV. Les équations de Diophante 119
Achat d’une tente 119
Vérification des comptes d’un magasin 124
Achat de timbres-poste 126
Achat de fruits 127
Deviner la date de naissance 129
La vente des poulets 131
Deux nombres et quatre opérations 133
Les côtés d’un rectangle 135
Deux nombres de deux chiffres 136
Les nombres de Pythagore 137
Equation indéterminée du troisième degré 142
Cent mille marks pour la démonstration d’un théorème 146
Chapitre V. La sixième opération mathématique 149
Quel est plus grand ? 151
Résoudre en un seul Coup d’œil 152
Paradoxes algébriques 153
Chapitre VI. Equations du second degré 157
Les poignées de main 157
Un essaim d’abeilles 158
La bande de singes 159
La prévoyance des équations 160
Le problème d’Euler 162
Les haut-parleurs 164
L’algèbre d’un voyage sur la Lune 166
« Un problème difficile » 169
Quels sont ces nombres? 172
Chapitre VIII. Maximums et minimums 173
Les deux trains 173
Où faut-il placer l’arrêt du train? 175
Comment tracer la route? 178
Maximum d’un produit 180
Minimum d’une somme 185
Une poutre de volume maximum 185
Les deux champs 186
Le cerf-volant 187
La construction d’une maison 189
Ün terrain pour une maison de campagne 191
Auge de section maximum 192
La capacité maximum d’un entonnoir 195
Le plus fort éclairage 196
Chapitre VIII. Progressions 199
Une progression très ancienne 199
L’algèbre sur le papier quadrillé 201
L’arrosage d’un jardin potager 202
La nourriture des poules 203
Une équipe de terrassiers 205
Les pommes 206
L’achat d’un cheval 207
La pension d’un soldat 208
Chapitre IX. La septième opération mathématique 210
Les concurrents des logarithmes 212
L’évolution des tables de logarithmes 213
Curiosités logarithmiques 214
Les logarithmes sur la scène 216
Les logarithmes à la ferme 219
Les logarithmes dans la musique 220
Les étoiles, le bruit et les logarithmes 222
Les logarithmes dans l’éclairage électrique 225
Testament pour des centaines d’années 226
Accroissement continu d’un capital 229
Le nombre « e » 230
Une astuce logarithmique 232
Ecrire un nombre quelconque à l’aide de trois 2 233
Posted in books, Editorial Mir, french, l’arithmétique, mathematics, mir books, popular science, Science Pour Tous
Tagged Equations du second degré, histoire des mathématiques, L'algèbre, Les équations de Diophante, Les nombres de Pythagore, logarithmes, mathématiques, Maximums et minimums, Nombres infinis, nombres premiers, problème d'Euler, Progressions, récréative, yakov perelman
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The Price Of Truth: The Story of Rare-Earth Elements – Trifonov
In this post, we will see the book The Price Of Truth: The Story of Rare-Earth Elements by D. N. Trifonov.
About the book
This book is about a fascinating family of chemical elements.
Popular science writers generally tend to overuse the word “fascinating” but in this case it gives a very apt description. Indeed, everything about these dozen and a half elements is fascinating-the histories of their discoveries, their properties and their applications.
Take Mendeleev’s periodic table of elements. The row under the table describes 14 elements from cerium to lutetium. When we add to them lanthanum, yttrium and scandium which are found in the third group of the table we have 17 elements comprising this fascinating family.
Seventeen? Was not it a dozen and a half just a few lines above? What is i t – a slip or carelessness?
Not at all. There is yet no complete agreement on which elements can be called the rare-earth elements. We shall discuss this problem below.
At last we gave the name to this family-the rare- earth elements belong to it. This rather interestingly sounding name contains two substantial errors.
Indeed, these elements are not all that rare.
Further studies demonstrated that the properties of these elements differed from those of the classic “earth” elements. Then their oxides should not be called earths.
We see that the name “rare earths” is just a relic of the past still used owing to a tradition. Sometimes it is better to keep an old familiar name, even if it has lost its former meaning, than invent a new one. Though this family of elements never lacked for names as we shall see in this story.
A prominent French chemist Georges Urbain once gave a very imaginative description of the history of rare-earth elements saying: “It was a sea of errors and the truth sank in it.”
But when the truth had been finally uncovered, when all intricacies of rare-earth elements had been studied it became clear that they were priceless, indeed. Rare earths make their contribution to the progress in practically all branches of modem technology.
Here we end this short but necessary introduction and begin our story.
About the author
Dmitri TRIFONOV is a Soviet chemist working on the history of chemistry. D. Trifonov was born in 1932. He graduated from the chemical department of Moscow University where he later was awarded the degree of doctor of science. Currently he is the head of a research group at the Institute of History of Natural Sciences and Technology of the USSR Academy of Sciences in Moscow .His research concerns mainly the theory and history of the periodic system of elements. He has written several books on the subject some of which have been translated into foreign languages. D. Trifonov is a well-known writer on science. He has written ten popular science books and a few dozen papers
The book was translated from Russian by O. Glebov and I. Poluyan and was published in 1987 by Mir Publishers.
You can get the book here.
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Contents
Short Introduction 6
The Thorny Path of Discoveries 9
The Birth of Truth 36
The Element from the “Sealed Envelope” 85
A Coming Age of Rare-Earth Elements? 111
In Conclusion: Are There “Relatives” of Rare-Earth Elements? 126
Posted in books, chemistry, discovery, history, mir books, mir publishers, popular science, science, soviet
Tagged Cerium, chemistry, discovery of elements, elements, history of chemistry, lanthanides, Lanthanum, moseley's law, Promethium, rare-earth elements, Scandium, uses of rare earth elements, Yttrium
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An Introduction To Computers – Sergeev, Vashkevich
In this post, we will see the book An Introduction To Computers by N. Sergeev and N. Vashkevich.
About the book
Because of the rapid growth and wide use of computers, in practically all fields of man’s activity, there is a need for students in all specialities to have a concise, yet thorough presentation of computer theory, design and operation.
This book is an attempt to fill this need. It covers the circuitry and operating principles of analog and digital computers, special-purpose computing devices, machines and systems. Ample space is devoted to a basic theory of similitude and simulation, and a fairly detailed description is given of basic functional elements, assemblies and units which make up typical analog and digital computers, in conjunction with an introduction to the mathematical and logical basis of electronic digital computers.
Separate sections deal with block-diagram synthesis of analog computers to solve algebraic, transcendental, ordinary and partial differential equations and their systems. The reader will find a survey of methods used to program problem solving on analog and digital computers, analog with a brief outline of digital differential analyzers and hybrid systems.
Written by leading authorities in the field of Soviet computer engineering with a wealth of teaching and research experience at colleges, the book is student-oriented in the presentation of the subject-matter.
The book will be of primary value to college students and faculty members and also to researchers and engineers concerned with the development and application of computers.The late PROFESSOR SERGEEV N.P., Doctor of Technical Sciences, was Head of the Chair (automatic information retrieval systems), rector of the Penza Polytechnic Institute. Lectured on computers (both digital and analog) and engineering and business applications of computers. Read lectures on automatic information retrieval systems. Has penned a total of 250 papers of which 200 have been published, including three monographs and 23 study aids on computer engineering and data processing.
PROFESSOR VASHKEVICH N.P., Doctor of Technical Sciences, Head of the Chair (computer engineering) at the Penza Polytechnic Institute.
Is a prominent authority in the Soviet Union on computation and computer engineering, especially in the field of product quality control.
Has penned 200 publications, including several study aids and a monograph.
Translated from the Russian by Boris Kuznetsov. First published 1976, revised from the 1973 Russian edition. Second printing in 1981 and third printing 1987
You can get the book here.
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Contents
Preface 8
Introduction 9
PART ONE. ANALOG COMPUTERS 11
Chapter I. Basic Theory of Similitude and Electric Modelling 11
1.1. Basic Conce pe of Similitude Theory 11
1.2. Synthesis of Electric Models for Physical Systems 18
1.3. Electric Modelling of Physical Fields 23
Chapter II. Computing Elements 34
2.1. General 34
2.2. Basic Elements of Computing Circuits 35
2.3. Summing Networks 54
2.4. Differentiators and Integrators 58
2.5. Function Generators 66
2.6. Function Multipliers-Dividers 85
Chapter III. Synthesis of Analog Differential Analyzers 97
3.1. Development of a Block Diagram for a Differential Analyzer 97
3.2. Main Units of Differential Analyzers 102
3.3. Problem Preparation for a Differential Analyzer 106
3.4. Problem Solving on an Electronic Differential Analyzer 113
Chapter IV. Algebraic and Transcendental Equation Solvers 118
Special-purpose purpose Algebraic Equation Solvers 118
Techniques for Solving Linear Algebraic Equations on Electronic Analog Computers 123
PART TWO. ELECTRONIC DIGITAL COMPUTERS
Chapter V. Mathematical Basis of Digital Computers 127
5.1. General 127
5.2. Arithmetic Principles 130
5.3. Switching Circuits and Boolean Algebra 142
Chapter VI. Basic Functional Elements of Electronic Digital Computers 156
6.1. Computer Elements 156
6.2. Logical Design 177
6.3 Logic Assemblies of Electronic Digital Computers 186
Chapter VII. Principal Units of an Electronic Digital Computer 200
7.1. Arithmetic Unit 200
7.2. Storage 204
7.3. Input/Output Equipment 227
7.4. Control Equipment 245
Chapter VIII. Principles of Programming 252
8.1. Basic Definitions. Hypothetical computer 252
8.2. Direct Programming 256
8.3. An Outline of FORTRAN Language 269
Chapter IX. Special-purpose Computers 287
9.1. Control Computers 287
9.2. Digital Differential Analyzers 289
9.3. Analog-Digital Computers 295
9.4. Prospects in the Development and Use of Electronic Computers 296
Bibliography 299
Index 300
Posted in books, computers, engineering, mir books, mir publishers, soviet, technology
Tagged analog computers, applications of computers, computer design, computer operation, computer programming, computer theory, computers, information retrieval systems, problem solving using computers, quality control, soviet, Soviet computer engineering, soviet technology, theory of similitude, theory of simulation
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