Problems in Mathematical Analysis – B. P. Demidovich – 2nd Edition


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 calculaiions. 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.

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  • Chapter I

    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

    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

    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
    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

    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.
    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.

    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.
    Sec. 1. Number Series 293
    Sec. 2. Functional Series 304
    Sec. 3. Taylor’s Series 318
    Sec. 4. Fourier’s Series 311

    Chapter IX

    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.

    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
  • Citation
    • Full Title: Problems in Mathematical Analysis
    • Author/s:
    • ISBN-10: 0846407612
    • Edition: 2nd Edition
    • Topic: Math
    • Subtopic: Mathematics
    • File Type: eBook
    • Idioma: English

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