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Book Source: Digital Library of India Item 2015.78471 dc.contributor.other: Whittaker And Company dc.date.accessioned: 2015-06-30T16:23:11Z dc.date.available: 2015-06-30T16:23:11Z dc.date.copyrightexpirydate: 0000-00-00 dc.date.digitalpublicationdate: 0000-00-00 dc.date.citation: 1898 dc.identifier: RMSC, IIIT-H dc.identifier.barcode: 2030020007063 dc.identifier.origpath: /data7/upload/0185/813 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/78471 dc.description.scanningcentre: RMSC, IIIT-H dc.description.slocation: OSU dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 510 dc.description.vendor: til dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Digital Library Of India dc.publisher: Mir Publishers dc.rights: Out_of_copyright dc.source.library: None dc.subject.classification: Natural Sciences dc.title: Problems In Mathematical Analysis dc.rights.holder: Macmillan And Co Limited

This study aid is based on many years’ experience of lecturing on mathematical analysis at the first course at the physics faculty of Moscow University. It is intended for students as well as for teachers, especially young ones, who are starting their lecturing career. The hook covers the analysis of functions of one variable, including the concepts of Lebesgue measure and Lebesgue integral. It is not a collection of problems in the ordinary sense. As can be seen from its structure, its aim is to help the student master the material both actively and informally. As a rule, the material in each section is divided into four subsections. Translated from the Russian by Irene Aleksanova

This study aid is based on many years’ experience of lecturing on mathematical analysis at the first course at the physics faculty of Moscow University. It is intended for students as well as for teachers, especially young ones, who are starting their lecturing career. The hook covers the analysis of functions of one variable, including the concepts of Lebesgue measure and Lebesgue integral. It is not a collection of problems in the ordinary sense. As can be seen from its structure, its aim is to help the student master the material both actively and informally. As a rule, the material in each section is divided into four subsections. Translated from the Russian by Irene Aleksanova

This study aid is based on many years’ experience of lecturing on mathematical analysis at the first course at the physics faculty of Moscow University. It is intended for students as well as for teachers, especially young ones, who are starting their lecturing career. The hook covers the analysis of functions of one variable, including the concepts of Lebesgue measure and Lebesgue integral. It is not a collection of problems in the ordinary sense. As can be seen from its structure, its aim is to help the student master the material both actively and informally. As a rule, the material in each section is divided into four subsections. Translated from the Russian by Irene Aleksanova

We now come to Problems in Mathematical Analysis edited by  B. P. Demidovich. The list of authors is G. Baranenkov,  B. Demidovich , V. Efimenko, S. Kogan , G. Lunts, E. Porshneva,  E. Sychera, S. Frolov, R. Shostak and A.  Yanpolsky . This collection of problems and exercises in mathematical analysis covers the maximum requirements of general courses in higher mathematics for higher technical schools. It contains over 3,000 problems sequentially arranged in Chapters I to X covering branches of higher mathematics (with the exception of analytical geometry) given in college courses. Particular attention is given to the most important sections of the course that require established skills (the finding of limits, differentiation techniques, the graphing of functions, integration techniques, the applications all of definite integrals, series, the solution of differential equations). Since some institutes have extended courses of mathematics, the authors have included problems on field theory, method, and the Fourier approximate calculations. Experience shows that problems given in this book not only fully satisfies the number of the requirements of the student, as far as practical mastering of the various sections of the course goes, but also enables the instructor to supply a varied choice of problems in each section to select problems for tests and examinations. Each chapter begins with a brief theoretical introduction that covers the basic definitions and formulas of that section of the course. Here the most important typical problems are worked out in full. We believe that this will greatly simplify the work of the student. Answers are given to all computational problems; one asterisk indicates that hints to the solution are given in the answers, two asterisks, that the solution is given. The are frequently illustrated by drawings. This collection of problems is the result of many years of teaching higher mathematics in the technical schools of the Soviet Union. It includes, in addition to original problems and examples, a large number of commonly used problems. This book was translated from the Russian by George Yankovsky . The  book was published by first Mir Publishers in 1970. All credits to the original uploader. Thanks Siddharth  for providing the link. PDF | OCR | 15.2 MB | Pages: 497 | Table of Contents Preface 9 Chapter I INTRODUCTION TO ANALYSIS  Sec. 1. Functions 11 Sec. 2. Graphs of Elementary Functions 16 Sec. 3 Limits 22 Sec. 4 Infinitely Small and Large Quantities 33 Sec. 5. Continuity of Functions 36 Chapter II DIFFERENTIATION OF FUNCTIONS Sec. 1. Calculating Derivatives Directly 42 Sec. 2. Tabular Differentiation 46 Sec. 3 The Derivatives of Functions Not Represented Explicitly 56 Sec. 4. Geometrical and Mechanical Applications of the Derivative 60 Sec. 5. Derivatives of Higher Orders 66 Sec. 6. Differentials of First and Higher Orders 71 Sec. 7. Mean Value Theorems 75 Sec. 8. Taylor's Formula 77 Sec. 9. The L'Hospital-Bernoulli Rule for Evaluating Indeterminate Forms 78 Chapter III THE EXTREMA OF A FUNCTION AND THE GEOMETRIC APPLICATIONS OF A DERIVATIVE Sec. 1. The Extrema of a Function of One Argument 83 Sec. 2. The Direction of Concavity. Points of Inflection 91 Sec. 3. Asymptotes 93 Sec. 4. Graphing Functions by Characteristic Points 96 Sec. 5. Differential of an Arc Curvature 101 Chapter IV INDEFINITE INTEGRALS Sec. 1. Direct Integration 107 Sec. 2. Integration by Substitution 113 Sec. 3. Integration by Parts 116 Sec. 4. Standard Integrals Containing a Quadratic Trinomial 118 Sec. 5. Integration of Rational Functions 121 Sec. 6. Integrating Certain Irrational Functions 125 Sec. 7. Integrating Trigoncrretric Functions 128 Sec. 8. Integration of Hyperbolic Functions 133 Sec. 9. Using Ingonometric and Hyperbolic Substitutions for Finding integrals of the Form $\int R(x, \sqrt{ax^2 + bx + c}) dx$ R Where R is a Rational Function Sec. 10. Integration of Various Transcendental Functions 135 Sec. 11. Using Reduction Formulas 135 Sec. 12. Miscellaneous Examples on Integration 136 Chapter V DEFINITE INTEGRALS Sec. 1. The Definite Integral as the Limit of a Sum 138 Sec. 2. Evaluating Definite Integrals by Means of Indefinite Integrals 140 Sec. 3 Improper Integrals 143 Sec. 4. Change of Variable in a Definite Integral 146 Sec. 5. Integration by Parts 149 Sec. 6. Mean-Value Theorem 150 Sec. 7. The Areas of Plane Figures 153 Sec 8. The Arc Length of a Curve 158 Sec 9 Volumes of Solids 161 Sec 10 The Area of a Surface of Revolution 166 Sec. 11. Moments. Centres of Gravity. Guldin's Theorems 168 Sec. 12. Applying Definite Integrals to the Solution of Physical Problems 173 Chapter VI. FUNCTIONS OF SEVERAL VARIABLES Sec. 1. Basic Notions 180 Sec. 2. Continuity 184 Sec. 3. Partial Derivatives 185 Sec. 4. Total Differential of a Function 187 Sec. 5. Differentiation of Composite Functions 190 Sec. 6. Derivative in a Given Direction and the Gradient of a Function 193 Sec. 7. Higher -Order Derivatives and Differentials 197 Sec. 8. Integration of Total Differentials 202 Sec. 9. Differentiation of Implicit Functions 205 Sec. 10. Change of Variables 211 Sec. 11. The Tangent Plane and the Normal to a Surface 217 Sec. 12. Taylor's Formula for a Function of Several Variables 220 Sec. 13. The Extremum of a Function of Several Variables 222 Sec. 14. Finding the Greatest and smallest Values of Functions 227 Sec. 15. Singular Points of Plane Curves 230 Sec. 16. Envelope 232 Sec. 17. Arc Length of a Space Curve 234 Sec. 18. The Vector Function of a Scalar Argument 235 Sec. 19. The Natural Trihedron of a Space Curve 238 Sec. 20. Curvature and Torsion of a Space Curve 242 Chapter VII. MULTIPLE AND LINE INTEGRALS Sec. 1. The Double Integral in Rectangular Coordinates 246 Sec. 2. Change of Variables in a Double Integral 252 Sec. 3. Computing Areas 256 Sec. 4. Computing Volumes 258 Sec. 5. Computing the Areas of Surfaces 259 Sec. 6 Applications of the Double Integral in Mechanics 260 Sec. 7. Triple Integrals 262 Sec. 8. Improper Integrals Dependent on a Parameter. Improper Multiple Integrals  269 Sec. 9. Line Integrals 273 Sec. 10. Surface Integrals 284 Sec. 11. The Ostrogradsky-Gauss Formula 286 Sec. 12. Fundamentals of Field Theory 288 Chapter VIII. SERIES Sec. 1. Number Series 293 Sec. 2. Functional Series 304 Sec. 3. Taylor's Series 318 Sec. 4. Fourier's Series 311 Chapter IX DIFFERENTIAL EQUATIONS Sec. 1. Verifying Solutions. Forming Differential Equations of Families of Curves. Initial Conditions 322 Sec. 2. First-Order Differential Equations 324 Sec. 3. First-Order Diflerential Equations with Variables Separable. Orthogonal Trajectories 327 Sec. 4. First-Order Homogeneous Differential Equations 330 Sec. 5. First-Order Linear Differential Equations. Bernoulli's Equation 332 Sec. 6 Exact Differential Equations. Integrating Factor 335 Sec 7 First-Order Differential Equations not Solved for the Derivative 337 Sec. 8. The Lagrange and Clairaut Equations 339 Sec. 9. Miscellaneous Exercises on First-Order Differential Equations 340 Sec. 10. Higher-Order Differential Equations 345 Sec. 11. Linear Differential Equations 349 Sec. 12. Linear Differential Equations of Second Order with Constant Coefficients 351 Sec. 13. Linear Differential Equations of Order Higher than Two with Constant Coefficients 356 Sec. 14. Euler's Equations 357 Sec. 15. Systems of Differential Equations 359 Sec. 16. Integration of Differential Equations by Means of Power Series 361 Sec. 17. Problems on Fourier's Method 363 Chapter X. APPROXIMATE CALCULATIONS Sec. 1. Operations on Approximate Numbers 367 Sec. 2. Interpolation of Functions 372 Sec. 3. Computing the Real Roots of Equations 376 Sec. 4. Numerical Integration of Functions 382 Sec. 5. Numerical Integration of Ordinary Differential Equations 384 Sec. 6. Approximating Fourier's Coefficients 393 ANSWERS 396 APPENDIX 475 I. Greek Alphabet 475 II. Some Constants 475 III. Inverse Quantities, Powers, Roots, Logarithms 476 IV. Trigonometric Functions 478 V. Exponential, Hyperbolic and Trigonometric Functions 479 VI. Some Curves 480

We now come to Problems in Mathematical Analysis edited by  B. P. Demidovich. The list of authors is G. Baranenkov,  B. Demidovich , V. Efimenko, S. Kogan , G. Lunts, E. Porshneva,  E. Sychera, S. Frolov, R. Shostak and A.  Yanpolsky . This collection of problems and exercises in mathematical analysis covers the maximum requirements of general courses in higher mathematics for higher technical schools. It contains over 3,000 problems sequentially arranged in Chapters I to X covering branches of higher mathematics (with the exception of analytical geometry) given in college courses. Particular attention is given to the most important sections of the course that require established skills (the finding of limits, differentiation techniques, the graphing of functions, integration techniques, the applications all of definite integrals, series, the solution of differential equations). Since some institutes have extended courses of mathematics, the authors have included problems on field theory, method, and the Fourier approximate calculations. Experience shows that problems given in this book not only fully satisfies the number of the requirements of the student, as far as practical mastering of the various sections of the course goes, but also enables the instructor to supply a varied choice of problems in each section to select problems for tests and examinations. Each chapter begins with a brief theoretical introduction that covers the basic definitions and formulas of that section of the course. Here the most important typical problems are worked out in full. We believe that this will greatly simplify the work of the student. Answers are given to all computational problems; one asterisk indicates that hints to the solution are given in the answers, two asterisks, that the solution is given. The are frequently illustrated by drawings. This collection of problems is the result of many years of teaching higher mathematics in the technical schools of the Soviet Union. It includes, in addition to original problems and examples, a large number of commonly used problems. This book was translated from the Russian by George Yankovsky . The  book was published by first Mir Publishers in 1970. All credits to the original uploader. Thanks Siddharth  for providing the link. PDF | OCR | 15.2 MB | Pages: 497 | Table of Contents Preface 9 Chapter I INTRODUCTION TO ANALYSIS  Sec. 1. Functions 11 Sec. 2. Graphs of Elementary Functions 16 Sec. 3 Limits 22 Sec. 4 Infinitely Small and Large Quantities 33 Sec. 5. Continuity of Functions 36 Chapter II DIFFERENTIATION OF FUNCTIONS Sec. 1. Calculating Derivatives Directly 42 Sec. 2. Tabular Differentiation 46 Sec. 3 The Derivatives of Functions Not Represented Explicitly 56 Sec. 4. Geometrical and Mechanical Applications of the Derivative 60 Sec. 5. Derivatives of Higher Orders 66 Sec. 6. Differentials of First and Higher Orders 71 Sec. 7. Mean Value Theorems 75 Sec. 8. Taylor's Formula 77 Sec. 9. The L'Hospital-Bernoulli Rule for Evaluating Indeterminate Forms 78 Chapter III THE EXTREMA OF A FUNCTION AND THE GEOMETRIC APPLICATIONS OF A DERIVATIVE Sec. 1. The Extrema of a Function of One Argument 83 Sec. 2. The Direction of Concavity. Points of Inflection 91 Sec. 3. Asymptotes 93 Sec. 4. Graphing Functions by Characteristic Points 96 Sec. 5. Differential of an Arc Curvature 101 Chapter IV INDEFINITE INTEGRALS Sec. 1. Direct Integration 107 Sec. 2. Integration by Substitution 113 Sec. 3. Integration by Parts 116 Sec. 4. Standard Integrals Containing a Quadratic Trinomial 118 Sec. 5. Integration of Rational Functions 121 Sec. 6. Integrating Certain Irrational Functions 125 Sec. 7. Integrating Trigoncrretric Functions 128 Sec. 8. Integration of Hyperbolic Functions 133 Sec. 9. Using Ingonometric and Hyperbolic Substitutions for Finding integrals of the Form $\int R(x, \sqrt{ax^2 + bx + c}) dx$ R Where R is a Rational Function Sec. 10. Integration of Various Transcendental Functions 135 Sec. 11. Using Reduction Formulas 135 Sec. 12. Miscellaneous Examples on Integration 136 Chapter V DEFINITE INTEGRALS Sec. 1. The Definite Integral as the Limit of a Sum 138 Sec. 2. Evaluating Definite Integrals by Means of Indefinite Integrals 140 Sec. 3 Improper Integrals 143 Sec. 4. Change of Variable in a Definite Integral 146 Sec. 5. Integration by Parts 149 Sec. 6. Mean-Value Theorem 150 Sec. 7. The Areas of Plane Figures 153 Sec 8. The Arc Length of a Curve 158 Sec 9 Volumes of Solids 161 Sec 10 The Area of a Surface of Revolution 166 Sec. 11. Moments. Centres of Gravity. Guldin's Theorems 168 Sec. 12. Applying Definite Integrals to the Solution of Physical Problems 173 Chapter VI. FUNCTIONS OF SEVERAL VARIABLES Sec. 1. Basic Notions 180 Sec. 2. Continuity 184 Sec. 3. Partial Derivatives 185 Sec. 4. Total Differential of a Function 187 Sec. 5. Differentiation of Composite Functions 190 Sec. 6. Derivative in a Given Direction and the Gradient of a Function 193 Sec. 7. Higher -Order Derivatives and Differentials 197 Sec. 8. Integration of Total Differentials 202 Sec. 9. Differentiation of Implicit Functions 205 Sec. 10. Change of Variables 211 Sec. 11. The Tangent Plane and the Normal to a Surface 217 Sec. 12. Taylor's Formula for a Function of Several Variables 220 Sec. 13. The Extremum of a Function of Several Variables 222 Sec. 14. Finding the Greatest and smallest Values of Functions 227 Sec. 15. Singular Points of Plane Curves 230 Sec. 16. Envelope 232 Sec. 17. Arc Length of a Space Curve 234 Sec. 18. The Vector Function of a Scalar Argument 235 Sec. 19. The Natural Trihedron of a Space Curve 238 Sec. 20. Curvature and Torsion of a Space Curve 242 Chapter VII. MULTIPLE AND LINE INTEGRALS Sec. 1. The Double Integral in Rectangular Coordinates 246 Sec. 2. Change of Variables in a Double Integral 252 Sec. 3. Computing Areas 256 Sec. 4. Computing Volumes 258 Sec. 5. Computing the Areas of Surfaces 259 Sec. 6 Applications of the Double Integral in Mechanics 260 Sec. 7. Triple Integrals 262 Sec. 8. Improper Integrals Dependent on a Parameter. Improper Multiple Integrals  269 Sec. 9. Line Integrals 273 Sec. 10. Surface Integrals 284 Sec. 11. The Ostrogradsky-Gauss Formula 286 Sec. 12. Fundamentals of Field Theory 288 Chapter VIII. SERIES Sec. 1. Number Series 293 Sec. 2. Functional Series 304 Sec. 3. Taylor's Series 318 Sec. 4. Fourier's Series 311 Chapter IX DIFFERENTIAL EQUATIONS Sec. 1. Verifying Solutions. Forming Differential Equations of Families of Curves. Initial Conditions 322 Sec. 2. First-Order Differential Equations 324 Sec. 3. First-Order Diflerential Equations with Variables Separable. Orthogonal Trajectories 327 Sec. 4. First-Order Homogeneous Differential Equations 330 Sec. 5. First-Order Linear Differential Equations. Bernoulli's Equation 332 Sec. 6 Exact Differential Equations. Integrating Factor 335 Sec 7 First-Order Differential Equations not Solved for the Derivative 337 Sec. 8. The Lagrange and Clairaut Equations 339 Sec. 9. Miscellaneous Exercises on First-Order Differential Equations 340 Sec. 10. Higher-Order Differential Equations 345 Sec. 11. Linear Differential Equations 349 Sec. 12. Linear Differential Equations of Second Order with Constant Coefficients 351 Sec. 13. Linear Differential Equations of Order Higher than Two with Constant Coefficients 356 Sec. 14. Euler's Equations 357 Sec. 15. Systems of Differential Equations 359 Sec. 16. Integration of Differential Equations by Means of Power Series 361 Sec. 17. Problems on Fourier's Method 363 Chapter X. APPROXIMATE CALCULATIONS Sec. 1. Operations on Approximate Numbers 367 Sec. 2. Interpolation of Functions 372 Sec. 3. Computing the Real Roots of Equations 376 Sec. 4. Numerical Integration of Functions 382 Sec. 5. Numerical Integration of Ordinary Differential Equations 384 Sec. 6. Approximating Fourier's Coefficients 393 ANSWERS 396 APPENDIX 475 I. Greek Alphabet 475 II. Some Constants 475 III. Inverse Quantities, Powers, Roots, Logarithms 476 IV. Trigonometric Functions 478 V. Exponential, Hyperbolic and Trigonometric Functions 479 VI. Some Curves 480

Ляшко И.И., Боярчук А.К., Гай Я.Г., Головач Г.П. - Математический анализ в примерах и задачах. Введение в анализ , производная, интеграл Ч. 1 , Вища школа , 1974 И.И. Ляшко, А.К. Боярчук, Я.Г. Гай, Г.П. Головач - Математический анализ в примерах и задачах. Часть 1 Mathematical analysis in examples and problems. INTRODUCTION TO ANALYSIS , DERIVATIVE , INTEGRAL Volume 1 "High School" , 1974 - I.I. Lyashko, A.K. Boyarchuk, Ya.G. Gai, G.P. Golovach Editorial Board of Literature on Mathematics and Physics Head of the Editorial Board - A.S. Makukha PUBLISHING ASSOCIATION SVISHCHA SHKOLA MAIN PUBLISHING HOUSE Kyiv — 1974  The manual consists of four chapters. At the beginning of each paragraph , the corresponding theoretical material is placed, and then examples and counterexamples are considered in detail.The book contains over 1400 examples and problems, to which detailed solutions are given.The manual is intended for students of the mechanical-mathematical and physical faculties, as well as the cybernetics faculties of universities, physical-mathematical faculties of pedagogical institutes and for students of technical universities. Fig. 158. This book is a textbook on mathematical analysis in its applied aspect for students of physics and mathematics and cybernetics at universities, as well as for students of pedagogical and technical universities. The textbook covers the sections of analysis studied in the first year: introduction to analysis, derivative, integral and their application.  The book contains over 1400 solved examples and problems, as well as over 150 graphs. The material for it was mainly problems and examples from the collection of B.P. Demidovich ; other sources were also used . When studying the course of mathematical analysis , undergraduate students usually encounter difficulties arising from the lack of necessary practice in solving problems, and later due to the large volume of information . The main goal of this book is to promote a deep assimilation of the theory, the development of concrete mathematical thinking of students, instilling in them the skills of solving examples and problems, understanding their physical essence . At the beginning of each paragraph, brief information on the theory is given in summary form. A large number of examples and problems are offered for independent solution.  This book will help the student master the methodology of applying theoretical material to solution of specific problems, acquire a creative approach to their solution. The manual will also be useful for teachers who conduct practical classes on mathematical analysis.  The authors express their deep gratitude to the doctor of physical and mathematical sciences, professor of the Kyiv Pedagogical Institute N.I. Shkil, associate professors of the Kyiv University M.I. Yadrenko , A. Ya. Dorogovtsev, V.N. Nagorny , V.A. Panasovich for a number of valuable comments that contributed to improving the content of the book.  Approved by the Ministry of Higher and Secondary Specialized Education of the Ukrainian SSR as a teaching aid for students of universities and technical higher educational institutions .  TABLE OF CONTENTS  Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . 3  Chapter I  Introduction to Analysis § 1. Real Numbers . . . . . . . . . . . . . . . . . . . . . . . 5 § 2. Sequence Theory . . . . . . . . . . . . . . . . . . . . .  15 § 3. The Concept of Function . . . . . . . . . . . . . . . . . . 51  § 4. Limit of Function . . . . . . . . . . . . . . . . . . . . . 61 § 5. Graphical representation of a function . . . . . . . . . .  116 § 6. Continuity of functions . . . . . . . . . . . . . . . . . . 146 § 7. Inverse function. Functions defined parametrically . ...   167 § 8. Uniform continuity of functions . . . . . . . . . . . . . 174 § 9. Functional equations . . . . . . . . . . . . . . . . . . . 182   Problems and examples for independent solution . . . . . . .  185 Chapter II Differential Calculus of Functions of One Variable § 1. Derivative of an Explicit Function . . . . . . . . . . . . . . . . . 192 § 2. Differential of a Function . . . . . . . . . . . . . . . . . . . . . 217 § 3. Derivative of the Inverse Function. Derivative of a Function   Given Parametrically , Derivative of a Function Given Implicitly . . . 223 § 4. Derivatives and Differentials of Higher Orders . . . . . . . . . . 228 § 5. Theorems of Rolle, Lagrange, Cauchy . . . . . . . . . . . . . . . 254 § 6. Increasing and decreasing functions. Inequalities . . . . . . . . 270 § 7. Direction of convexity of the function graph. Inflection points . . 285  § 8. Disclosure of uncertainties . . . . . . . . . . . . . . . . . . . 291 § 9. Taylor's formula . . . . . . . . . . . . . . . . . . . . . . . . . 302 § 10. Extremum of a function. Largest and smallest values of a function . 316 § 11. Plotting function graphs by characteristic points . . . . . . . . 330 § 12. Problems on the maximum and minimum of a function . . . . . . . . 347   Problems and examples for independent solution . . . . . . . . . . 354 Chapter III Indefinite integral § 1. Simplest indefinite integrals . . . . . . . . . . . . . . . 363 § 2. Integration of rational functions . . . . . . . . . . . . . . 392 § 3. Integration of irrational functions . . . . . . . . . . . . . 413 § 4. Integration of trigonometric functions . . . . . . . . . . . . 430 § 5. Integration of various transcendental functions . . . . . . . 445 § 6. Various examples on integration of functions . . . . . . . . . 455   Tasks and examples for independent solution . . . . . . . . . . . 467 Chapter IV Definite Integral § 1. Definite Integral as Limit of a Sum  . . . . . . . . . . 470 § 2. Calculating Definite Integrals with Indefinite  . . . .  492 § 3. Mean Theorems . . . . . . . . . . . . . . . . . . . . .  535 § 4. Improper Integrals . . . . . . . . . . . . . . . . . . . 546 § 5. Calculating Areas . . . . . . . . . . . . . . . . . . . 576 § 6. Calculating Arc Lengths . . . . . . . . . . . . . . . . 591  § 7. Calculation of volumes . . . . . . . . . . . . . . . .  601 § 8. Calculation of areas of surfaces of revolution . . . .  621 § 9.  General scheme of application of definite integral.  Calculation of moment , coordinates of the center of gravity . . . 632 § 10. Problems from mechanics and physics . ... . . . . 645 § 11. Approximate calculation of definite integrals . . . . . . . 654  Problems and examples for independent solution . . . . . . . . . 664 Answers Chapter I . . . . . . . . . . . . . . . . . 670 Chapter II  . . . . . . . . . . . . . . . . 671 Chapter III  . . . . . . . . . . . . . . . . 674 Chapter IV  . . . . . . . . . . . . . . . .  676

Ляшко И.И., Боярчук А.К., Гай Я.Г., Головач Г.П. - Математический анализ в примерах и задачах. Введение в анализ , производная, интеграл Ч. 1 , Вища школа , 1974 И.И. Ляшко, А.К. Боярчук, Я.Г. Гай, Г.П. Головач - Математический анализ в примерах и задачах. Часть 1 Mathematical analysis in examples and problems. INTRODUCTION TO ANALYSIS , DERIVATIVE , INTEGRAL Volume 1 "High School" , 1974 - I.I. Lyashko, A.K. Boyarchuk, Ya.G. Gai, G.P. Golovach Editorial Board of Literature on Mathematics and Physics Head of the Editorial Board - A.S. Makukha PUBLISHING ASSOCIATION SVISHCHA SHKOLA MAIN PUBLISHING HOUSE Kyiv — 1974  The manual consists of four chapters. At the beginning of each paragraph , the corresponding theoretical material is placed, and then examples and counterexamples are considered in detail.The book contains over 1400 examples and problems, to which detailed solutions are given.The manual is intended for students of the mechanical-mathematical and physical faculties, as well as the cybernetics faculties of universities, physical-mathematical faculties of pedagogical institutes and for students of technical universities. Fig. 158. This book is a textbook on mathematical analysis in its applied aspect for students of physics and mathematics and cybernetics at universities, as well as for students of pedagogical and technical universities. The textbook covers the sections of analysis studied in the first year: introduction to analysis, derivative, integral and their application.  The book contains over 1400 solved examples and problems, as well as over 150 graphs. The material for it was mainly problems and examples from the collection of B.P. Demidovich ; other sources were also used . When studying the course of mathematical analysis , undergraduate students usually encounter difficulties arising from the lack of necessary practice in solving problems, and later due to the large volume of information . The main goal of this book is to promote a deep assimilation of the theory, the development of concrete mathematical thinking of students, instilling in them the skills of solving examples and problems, understanding their physical essence . At the beginning of each paragraph, brief information on the theory is given in summary form. A large number of examples and problems are offered for independent solution.  This book will help the student master the methodology of applying theoretical material to solution of specific problems, acquire a creative approach to their solution. The manual will also be useful for teachers who conduct practical classes on mathematical analysis.  The authors express their deep gratitude to the doctor of physical and mathematical sciences, professor of the Kyiv Pedagogical Institute N.I. Shkil, associate professors of the Kyiv University M.I. Yadrenko , A. Ya. Dorogovtsev, V.N. Nagorny , V.A. Panasovich for a number of valuable comments that contributed to improving the content of the book.  Approved by the Ministry of Higher and Secondary Specialized Education of the Ukrainian SSR as a teaching aid for students of universities and technical higher educational institutions .  TABLE OF CONTENTS  Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . 3  Chapter I  Introduction to Analysis § 1. Real Numbers . . . . . . . . . . . . . . . . . . . . . . . 5 § 2. Sequence Theory . . . . . . . . . . . . . . . . . . . . .  15 § 3. The Concept of Function . . . . . . . . . . . . . . . . . . 51  § 4. Limit of Function . . . . . . . . . . . . . . . . . . . . . 61 § 5. Graphical representation of a function . . . . . . . . . .  116 § 6. Continuity of functions . . . . . . . . . . . . . . . . . . 146 § 7. Inverse function. Functions defined parametrically . ...   167 § 8. Uniform continuity of functions . . . . . . . . . . . . . 174 § 9. Functional equations . . . . . . . . . . . . . . . . . . . 182   Problems and examples for independent solution . . . . . . .  185 Chapter II Differential Calculus of Functions of One Variable § 1. Derivative of an Explicit Function . . . . . . . . . . . . . . . . . 192 § 2. Differential of a Function . . . . . . . . . . . . . . . . . . . . . 217 § 3. Derivative of the Inverse Function. Derivative of a Function   Given Parametrically , Derivative of a Function Given Implicitly . . . 223 § 4. Derivatives and Differentials of Higher Orders . . . . . . . . . . 228 § 5. Theorems of Rolle, Lagrange, Cauchy . . . . . . . . . . . . . . . 254 § 6. Increasing and decreasing functions. Inequalities . . . . . . . . 270 § 7. Direction of convexity of the function graph. Inflection points . . 285  § 8. Disclosure of uncertainties . . . . . . . . . . . . . . . . . . . 291 § 9. Taylor's formula . . . . . . . . . . . . . . . . . . . . . . . . . 302 § 10. Extremum of a function. Largest and smallest values of a function . 316 § 11. Plotting function graphs by characteristic points . . . . . . . . 330 § 12. Problems on the maximum and minimum of a function . . . . . . . . 347   Problems and examples for independent solution . . . . . . . . . . 354 Chapter III Indefinite integral § 1. Simplest indefinite integrals . . . . . . . . . . . . . . . 363 § 2. Integration of rational functions . . . . . . . . . . . . . . 392 § 3. Integration of irrational functions . . . . . . . . . . . . . 413 § 4. Integration of trigonometric functions . . . . . . . . . . . . 430 § 5. Integration of various transcendental functions . . . . . . . 445 § 6. Various examples on integration of functions . . . . . . . . . 455   Tasks and examples for independent solution . . . . . . . . . . . 467 Chapter IV Definite Integral § 1. Definite Integral as Limit of a Sum  . . . . . . . . . . 470 § 2. Calculating Definite Integrals with Indefinite  . . . .  492 § 3. Mean Theorems . . . . . . . . . . . . . . . . . . . . .  535 § 4. Improper Integrals . . . . . . . . . . . . . . . . . . . 546 § 5. Calculating Areas . . . . . . . . . . . . . . . . . . . 576 § 6. Calculating Arc Lengths . . . . . . . . . . . . . . . . 591  § 7. Calculation of volumes . . . . . . . . . . . . . . . .  601 § 8. Calculation of areas of surfaces of revolution . . . .  621 § 9.  General scheme of application of definite integral.  Calculation of moment , coordinates of the center of gravity . . . 632 § 10. Problems from mechanics and physics . ... . . . . 645 § 11. Approximate calculation of definite integrals . . . . . . . 654  Problems and examples for independent solution . . . . . . . . . 664 Answers Chapter I . . . . . . . . . . . . . . . . . 670 Chapter II  . . . . . . . . . . . . . . . . 671 Chapter III  . . . . . . . . . . . . . . . . 674 Chapter IV  . . . . . . . . . . . . . . . .  676

Problems in Mathematical Analysis Demidovich

Problems in Mathematical Analysis Demidovich

Problems in Mathematical Analysis Demidovich

We consider the design and analysis of numerical methods for approximating positive solutions to nonlinear geometric elliptic partial differential equations containing critical exponents. This class of problems includes the Yamabe problem and the Einstein constraint equations, which simultaneously contain several challenging features: high spatial dimension n >= 3, varying (potentially non-smooth) coefficients, critical (even super-critical) nonlinearity, non-monotone nonlinearity (arising from a non-convex energy), and spatial domains that are typically Riemannian manifolds rather than simply open sets in Rn. These problems may exhibit multiple solutions, although only positive solutions typically have meaning. This creates additional complexities in both the theory and numerical treatment of such problems, as this feature introduces both non-uniqueness as well as the need to incorporate an inequality constraint into the formulation. In this work, we consider numerical methods based on Galerkin-type discretization, covering any standard bases construction (finite element, spectral, or wavelet), and the combination of a barrier method for nonconvex optimization and global inexact Newton-type methods for dealing with nonconvexity and the presence of inequality constraints. We first give an overview of barrier methods in non-convex optimization, and then develop and analyze both a primal barrier energy method for this class of problems. We then consider a sequence of numerical experiments using this type of barrier method, based on a particular Galerkin method, namely the piecewise linear finite element method, leverage the FETK modeling package. We illustrate the behavior of the primal barrier energy method for several examples, including the Yamabe problem and the Hamiltonian constraint.

We consider the design and analysis of numerical methods for approximating positive solutions to nonlinear geometric elliptic partial differential equations containing critical exponents. This class of problems includes the Yamabe problem and the Einstein constraint equations, which simultaneously contain several challenging features: high spatial dimension n >= 3, varying (potentially non-smooth) coefficients, critical (even super-critical) nonlinearity, non-monotone nonlinearity (arising from a non-convex energy), and spatial domains that are typically Riemannian manifolds rather than simply open sets in Rn. These problems may exhibit multiple solutions, although only positive solutions typically have meaning. This creates additional complexities in both the theory and numerical treatment of such problems, as this feature introduces both non-uniqueness as well as the need to incorporate an inequality constraint into the formulation. In this work, we consider numerical methods based on Galerkin-type discretization, covering any standard bases construction (finite element, spectral, or wavelet), and the combination of a barrier method for nonconvex optimization and global inexact Newton-type methods for dealing with nonconvexity and the presence of inequality constraints. We first give an overview of barrier methods in non-convex optimization, and then develop and analyze both a primal barrier energy method for this class of problems. We then consider a sequence of numerical experiments using this type of barrier method, based on a particular Galerkin method, namely the piecewise linear finite element method, leverage the FETK modeling package. We illustrate the behavior of the primal barrier energy method for several examples, including the Yamabe problem and the Hamiltonian constraint.

Математический анализ в примерах и задачах. В 2-х частях Часть 2 «Вища школа», Часть 2, 1977 Ляшко И.И., Боярчук А.К., Гай Я.Г., Головач Г.П.- Математический анализ в примерах и задачах. Ряды, функций нескольких переменных, кратные и криволинейные интегралы;;Ч. 2;Вища школа;1977 И.И. Ляшко, А.К. Боярчук, Я.Г. Гай, Г.П. Головач;Математический анализ в примерах и задачах. Часть 2;«Вища школа»;1977 Lyashko I.I., Boyarchuk A.K., Gai Ya.G., Golovach G.P.;Mathematical analysis in examples and problems. Series, functions of several variables, multiple and curvilinear integrals - Volume 2 ; Higher school - 1977 Volume 2. Series, Functions of Several Variables, Multiple and Curvilinear Integrals. The manual consists of four chapters. Each section begins with the relevant theoretical material, followed by a detailed examination of examples and counterexamples. It contains over 1,140 solved examples and problems, as well as examples and problems for independent solution. The manual is intended for students of the faculties of mechanics and mathematics, physics, and cybernetics of universities, faculties of physics and mathematics of pedagogical institutes, and for students of technical universities. Contents Chapter  Series § 1. Numerical series. Tests for convergence of series of constant sign ......Page 4 § 2. Tests for convergence of series of alternating sign ......Page 33 § 3. Operations on series ......Page 52 § 4. Functional sequences and series. Properties of uniformly converging functional sequences and series ......Page 54 § 5. Power series ......Page 82 § 6. Fourier series ......Page 118 § 7. Summation of series ......Page 142 § 8. Finding definite integrals using series ......Page 158 Problems and examples for independent solution ......Page 166 Chapter  Functions of Several Variables § 1. Limit of a function. Continuity ......Page 171 § 2. Partial derivatives. Differential of a function ......Page 184 § 3. Metric spaces ......Page 208 § 4. Implicit functions ......Page 218 § 5. Change of variables ......Page 243 § 6. Taylor's formula. Some geometric applications of differential calculus ......Page 277 § 7. Extrema of functions of several variables ......Page 296 Problems and examples for independent solution ......Page 333 Chapter Improper Integrals § 1. Proper integrals depending on a parameter ......Page 337 § 2. Improper integrals depending on a parameter. Uniform convergence of integrals ......Page 355 § 3. Differentiation and integration of improper integrals under the integral sign ......Page 380 § 4. Euler integrals ......Page 404 § 5. Fourier integral formula ......Page 418 Problems and examples for independent solution ......Page 424 Chapter Multiple and Curvilinear Integrals § 1. Riemann integral on a compact set. Double integrals ......Page 428 § 2. Calculating areas using double integrals ......Page 460 § 3. Calculating volumes using double integrals ......Page 473 § 4. Calculating surface areas using double integrals ......Page 483 § 5. Applications of double integrals to solving problems in mechanics ......Page 494 § 6. Triple integrals ......Page 511 § 7. Calculating volumes using triple integrals ......Page 519 § 8. Applications of triple integrals to solving problems in mechanics ......Page 536 § 9. Curvilinear integrals ......Page 559 § 10. Green's formula ......Page 593 § 11. Physical applications of curvilinear integrals ......Page 606 § 12. Surface integrals ......Page 615 § 13. Stokes formula ......Page 637 § 14. Ostrogradsky formula ......Page 641 § 15. Elements of vector analysis ......Page 649 Problems and examples for independent solution ......Page 664 Chapter I ...... 667 Chapter II ......Page 667 Chapter III ...... 668 Chapter IV ......Page 668

Математический анализ в примерах и задачах. В 2-х частях Часть 2 «Вища школа», Часть 2, 1977 Ляшко И.И., Боярчук А.К., Гай Я.Г., Головач Г.П.- Математический анализ в примерах и задачах. Ряды, функций нескольких переменных, кратные и криволинейные интегралы;;Ч. 2;Вища школа;1977 И.И. Ляшко, А.К. Боярчук, Я.Г. Гай, Г.П. Головач;Математический анализ в примерах и задачах. Часть 2;«Вища школа»;1977 Lyashko I.I., Boyarchuk A.K., Gai Ya.G., Golovach G.P.;Mathematical analysis in examples and problems. Series, functions of several variables, multiple and curvilinear integrals - Volume 2 ; Higher school - 1977 Volume 2. Series, Functions of Several Variables, Multiple and Curvilinear Integrals. The manual consists of four chapters. Each section begins with the relevant theoretical material, followed by a detailed examination of examples and counterexamples. It contains over 1,140 solved examples and problems, as well as examples and problems for independent solution. The manual is intended for students of the faculties of mechanics and mathematics, physics, and cybernetics of universities, faculties of physics and mathematics of pedagogical institutes, and for students of technical universities. Contents Chapter  Series § 1. Numerical series. Tests for convergence of series of constant sign ......Page 4 § 2. Tests for convergence of series of alternating sign ......Page 33 § 3. Operations on series ......Page 52 § 4. Functional sequences and series. Properties of uniformly converging functional sequences and series ......Page 54 § 5. Power series ......Page 82 § 6. Fourier series ......Page 118 § 7. Summation of series ......Page 142 § 8. Finding definite integrals using series ......Page 158 Problems and examples for independent solution ......Page 166 Chapter  Functions of Several Variables § 1. Limit of a function. Continuity ......Page 171 § 2. Partial derivatives. Differential of a function ......Page 184 § 3. Metric spaces ......Page 208 § 4. Implicit functions ......Page 218 § 5. Change of variables ......Page 243 § 6. Taylor's formula. Some geometric applications of differential calculus ......Page 277 § 7. Extrema of functions of several variables ......Page 296 Problems and examples for independent solution ......Page 333 Chapter Improper Integrals § 1. Proper integrals depending on a parameter ......Page 337 § 2. Improper integrals depending on a parameter. Uniform convergence of integrals ......Page 355 § 3. Differentiation and integration of improper integrals under the integral sign ......Page 380 § 4. Euler integrals ......Page 404 § 5. Fourier integral formula ......Page 418 Problems and examples for independent solution ......Page 424 Chapter Multiple and Curvilinear Integrals § 1. Riemann integral on a compact set. Double integrals ......Page 428 § 2. Calculating areas using double integrals ......Page 460 § 3. Calculating volumes using double integrals ......Page 473 § 4. Calculating surface areas using double integrals ......Page 483 § 5. Applications of double integrals to solving problems in mechanics ......Page 494 § 6. Triple integrals ......Page 511 § 7. Calculating volumes using triple integrals ......Page 519 § 8. Applications of triple integrals to solving problems in mechanics ......Page 536 § 9. Curvilinear integrals ......Page 559 § 10. Green's formula ......Page 593 § 11. Physical applications of curvilinear integrals ......Page 606 § 12. Surface integrals ......Page 615 § 13. Stokes formula ......Page 637 § 14. Ostrogradsky formula ......Page 641 § 15. Elements of vector analysis ......Page 649 Problems and examples for independent solution ......Page 664 Chapter I ...... 667 Chapter II ......Page 667 Chapter III ...... 668 Chapter IV ......Page 668

A fairly detailed study of many existing theories of damage was conducted. One conclusion, perhaps not surprising, is that there is poor agreement among researchers in this field as to what constitutes a physically correct measure of damage in both brittle and ductile materials. For anisotropic damage, scalar-, vector-, and tensor-valued damage variables have been proposed for materials that undergo elastic and elastoplastic deformation. Numerous deficiencies and inconsistencies, both physical and mathematical, exist in some of the more publicized theories. In general, the field is still quite immature and the general acceptance of basic principles and definition of terms have neither the experimental support nor the consensus of workers in the field to form the nucleus of a general mathematical theory. The field of isotropic damage is in somewhat better shape. In the present study, a critical look at the subject was conducted and several new results were produced.

A fairly detailed study of many existing theories of damage was conducted. One conclusion, perhaps not surprising, is that there is poor agreement among researchers in this field as to what constitutes a physically correct measure of damage in both brittle and ductile materials. For anisotropic damage, scalar-, vector-, and tensor-valued damage variables have been proposed for materials that undergo elastic and elastoplastic deformation. Numerous deficiencies and inconsistencies, both physical and mathematical, exist in some of the more publicized theories. In general, the field is still quite immature and the general acceptance of basic principles and definition of terms have neither the experimental support nor the consensus of workers in the field to form the nucleus of a general mathematical theory. The field of isotropic damage is in somewhat better shape. In the present study, a critical look at the subject was conducted and several new results were produced.

The research program being funded here emphasizes problems in turbulence and turbulent diffusion which are inherently statistical and involve many spatio-temporal scales. One goal of the research is to achieve a better theoretical understanding of turbulent (reaction) diffusion which is crucial for many applications in environmental science and engineering such as the tracking of pollutants in the atmosphere, the behavior of chemical tracers in the ocean and porous media, and turbulent combustion. Other parts of the research emphasize the interaction and generation of both small scales and large scale coherent structure in various anisotropic turbulent flows from both a statistical and deterministic point of view. The approach to all of these issues involves a combination of asymptotic analysis, numerical computation, and theoretical analysis to gain insight into these complex and important phenomena.

B. P. Demidovich - Problems In Mathematical Analysis, MIR Publishers, Moscow

-real numbers; limit of a sequence; limit of a function, continuity of function; derivatives and differentials; the indefinite integral; fundamental theorems on continuous and differentiable functions; investigating the behaviour of a function and constructing graphs; the definite integral; lebesgue measure and lebesgue integral; answers and hints-