Problems in Mathematical Analysis

Title Page
CONTENTS
PREFACE
Chapter I INTRODUCTION TO ANALYSIS
Sec. 1. Functlcns
Sec. 2. Graphs of Elementary Functions
Sec. 3. Limits
Sec. 4. Infinitely Small and Large Quantities
Sec. 5. Continuity of Functions
Chapter II DIFFERENTIATION OF FUNCTIONS
Sec. 1. Calculating Derivatives Directly
Sec. 2. Tabular Differentiation
Sec. 3. The Derivatives of Functions Not Represented Explicitly
Sec. 4. Geometrical and Mechanical Applications of the Derivative
Sec. 5. Derivatives of Higher Orders
Sec. 6. Differentials of First and Higher Orders
Sec. 7. Mean-Value Theorems
Sec. 8. Taylor's Formula
Sec. 9. The L'Hospitai-Bernoulli Rule for Evaluating Indeterminate Forms
Chapter Ill THE EXTREMA OF A FUNCTION AND THE GEOMETRIC APPLICATIONS OF A DERIVATIVE
Sec. 1. The Extrema of a Function of One Argument
Sec. 2. The Direction of Concavity. Points of Inflection
Sec. 3. Asymptotes
Sec. 4. Graphing Functions by Characteristic Points
Sec. 5. Differential of an Arc. Curvature
Chapter IV INDEFINITE INTEGRALS
Sec. 1. Direct Integration
Sec. 2. Integration by Substitution
Sec. 3. lntegration by Parts
Sec. 4. Standard Integrals Containing a Quadratic Trinomial
Sec. 5. Integration of Rational Functions
Sec. 6. lntegrating Certain Irrational Functions
Sec. 7. Integrating Trigonometric Functions
Sec. 8. Integration of Hyperbolic Functions
Sec. 9. Using Trigonometric and Hyperbolic Substitutions for Finding Integrals of the Form $\int R (x, \sqrt{ax^[2}+bx+c) dx$
Sec. 10. Integration of Various Transcendental Functions
Sec. 11. Using Reduction Formulas
Sec. 12. Miscellaneous Examples on Integration
Chapter V DEFINITE INTEGRALS
Sec. 1. The Definite Integral as the Limit of a Sum
Sec. 2. Evaluating Definite Integrals by Means of Indefinite Integrals
Sec. 3. Improper Integrals
Sec. 4. Change of Variable in a Definite Integral
Sec. 5. Integration by Parts
Sec. 6. Mean-Value Theorem
Sec. 7. The Areas of Plane Figures
Sec. 8. The Arc Length of a Curve
Sec. 9. Volumes of Solids
Sec. 10. The Area of a Surface of Revolution
Sec. 11. Moments. Centres of Gravity. Guldin's Theorems
Sec. 12. Applying Definite Integrals to the Solution of Physical Problems
Chapter VI FUNCTIONS OF SEVERAL VARIABLES
Sec. 1. Basic Notions
Sec. 2. Continuity
Sec. 3. Partial Derivatives
Sec. 4. Total Differential of a Function
Sec. 5. Differentiation of Composite Functions
Sec. 6. Derivative in a Given Direction and the Gradient of a Function
Sec. 7. Higher-Order Derivatives and Differentials
Sec. 8. Integration of Total Differentials
Sec. 9. Differentiation of Implicit Functions
Sec. 10. Change of Variables
Sec. 11. The Tangent Plane and the Normal to a Surface
Sec. 12. Taylor's Formula for a Function of Several Variables
Sec. 13. The Extremum of a Function of Several Variables
Sec. 14. Finding the Greatest and Smallest Values of Functions
Sec. 15. Singular Points of Plane Curves
Sec. 16. Envelope
Sec. 17. Arc Length of a Space Curve
Sec. 18. The Vector Function of a Scalar Argument
Sec. 19. The Natural Trihedron of a Space Curve
Sec. 20. Curvature and Torsion of a Space Curve
Chapter VII MULTIPLE AND LINE INTEGRALS
Sec. 1. The Double Integral in Rectangular Coordinates
Sec. 2. Change of Variables in a Double Integral
Sec. 3. Computing Areas
Sec. 4. Computing Volumes
Sec. 5. Computing the Areas of Surfaces
Sec. 6. Applications of the Double Integral in Mechanics
Sec. 7. Triple Integrals
Sec. 8. Improper Integrals Dependent on a Parameter. Improper Multiple Integrals
Sec. 9. Line Integrals
Sec. 10. Surface Integrals
Sec. 11. The Ostrogradsky-Gauss Formula
Sec. 12. Fundamentals of Field Theory
Chapter V Ill SERIES
Sec. 1. Number Series
Sec. 2. Functional Series
Sec. 3. Taylor's Series
Sec. 4. Fourier Series
Chapter IX DlFFERENTIAL EQUATIONS
Sec. 1. Verifying Solutions. Forming Differential Equations of Families of Curves. Initial Conditions
Sec. 2. First-Order Differential Equations
Sec. 3. First-Order Differential Equations with Variables Separable. Orthogonal Trajectories
Sec. 4. First-Order Homogeneous Differential Equations
Sec. 5. First-Order Linear Differential Equations. Bernoulli's Equation
Sec. 6. Exact Differential Equations. Integrating Factor
Sec. 7. First-Order Differential Equations not Solved for the Derivative
Sec. 8. The Lagrange and Clairaut Equations
Sec. 9. Miscellaneous Exercises on First-Order Differential Equations
Sec. 10. Higher-Order Differential Equations
Sec. 11. Linear Differential Equations
Sec. 12. Linear Differential Equations of Second Order with Constant Coefficients
Sec. 13. Linear Differential Equations of Order Higher than Two with Constant Coefficients
Sec. 14. Euler's Equations
Sec. 15. Systems of Differential Equations
Sec. 16. Integration of Differential Equations by Means of Power Series
Sec. 17. Problems on Fourier's Method
Chapter X APPROXIMATE CALCULATIONS
Sec. 1. Operations on Approximate Numbers
Sec. 2. Interpolation of Functions
Sec. 3. Computing the Real Roots of Equations
Sec. 4. Numerical Integration of Functions
Sec. 5. Numerical Integration of Ordinary Differential Equations
Sec. 6. Approximating Fourier Coefficients
ANSWERS
Chapter I
Chapter II
Chapter Ill
Chapter IV
Chapter V
Chapter VI
Chapter VII
Chapter VIII
Chapter IX
Chapter X
APPENDIX
I. Greek Alphabet
II. Some Constants
Ill. Inverse Quantities, Powers, Roots, Logarithms
IV. Trigonometric Functions
V. Exponential, Hyperbolic and Trigonometric Functions
VI. Some Curves (for Reference)
INDEX