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