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

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