ISBN: Ostalo
Godina izdanja: 1965
Jezik: Engleski
Oblast: Matematika
Autor: Strani

B. Demidovich - Problems in Mathematical Analysis
Mir Publishers, 1965
496 tvrdi povez
stanje: dobro-, predlist je otcepljen, beleške grafitnom olovkom.

G. Yankovsky (Translator)

PEACE PUBLISHERS, Moscow
BORIS PAVLOVICH DEMIDOVICH
with G. Baranenkov

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 all 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 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, the Fourier method, and approximate calculations. Experience shows that the number of problems given in this book not only fully satisfies 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 and 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
problems 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 exam-
ples, a large number of commonly used problems.

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 Derivates 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 f R (x, ^a^ + bx + c) dx, Where R is a Rational Function 133
Sec 10 Integration of Vanou* 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 Ccfirite Integrals by Means of Indefinite Integrals 140
Sec. 3 Improper Integrals 143
Sec 4 Charge 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 torrents 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 HigKei -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 Firdirg the Greatest and tallest Values of Functions . . 227
Sec 15 Smcular Points of Plane Curves 230
Sec 16 Envelope . . 232
Sec. 17. Arc Length o! 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 230
Sec. 7. Triple Integrals 262
Sec. 8. Improper Integrals Dependent on a Parameter. Improper Multifle 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 311
Sec. 4. Fourier`s Series 318
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 Differential 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^Rcal Roots of Equations 376
Sec. 4 Numerical, Integration of Functions 382
Sec. 5. Nun er:ca1 Integration of Ordinary DilUrtntial Equations . . 384
Sec. 6. Approximating Ftuncr`s Coefficients 3>3
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


Nonfiction, Mathematics