A Collection of Problems on a Course of Mathematical Analysis
International Series of Monographs in Pure and Applied Mathematics
Published on 6. June 2016
602 pages
978-1-4831-3734-6 (ISBN)
Description
A Collection of Problems on a Course of Mathematical Analysis is a collection of systematically selected problems and exercises (with corresponding solutions) in mathematical analysis. A common instruction precedes a group of problems of the same type. Problems with a physics content are preceded by the necessary physical laws. In the case of more or less difficult problems, hints are given in the answers.
This book is comprised of 15 chapters and begins with an overview of functions and methods of specifying them; notation for and classification of functions; elementary investigation of functions; and trigonometric and inverse trigonometric functions. The following chapters deal with limits and tests for their existence; differential calculus, with emphasis on derivatives and differentials; functions and curves; definite and indefinite integrals; and methods of evaluating definite integrals. Some applications of the integral in geometry, statics, and physics are also considered; along with functions of several variables; multiple integrals and iterated integration; line and surface integrals; and differential equations. The final chapter is devoted to trigonometric series.
This monograph is intended for students studying mathematical analysis within the framework of a technical college course.
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Content
Foreword to the Tenth Russian EditionI. Functions 1. Functions and Methods of Specifying Them 2. Notation for and Classification of Functions 3. Elementary Investigation of Functions 4. Elementary Functions 5. The Inverse Functions. Power, Exponential and Logarithmic Functions 6. The Trigonometric and Inverse Trigonometric Functions 7. Numerical ProblemsII. Limits 1. Basic Definitions 2. Orders of Magnitude. Tests for the Existence of a Limit 3. Continuous Functions 4. Finding Limits. Comparison of InfinitesimalsIII. Derivatives and Differentials. Differential Calculus 1. Derivatives. The Rate of Change of a Function 2. Differentiation of Functions 3. Differentials. Differentiability of a Function 4. Derivative as Rate of Change (Further Examples) 5. Repeated DifferentiationIV. The Investigation of Functions and Curves 1. The Behavior of a Function "at a Point" 2. Applications of the First Derivative 3. Applications of the Second Derivative 4. Auxiliary Problems. Solution of Equations 5. Taylor's Formula and its Applications 6. Curvature 7. Numerical ProblemsV. The Definite Integral 1. The Definite Integral and its Elementary Properties 2. Fundamental Properties of the Definite IntegralVI. The Indefinite Integral. Integral Calculus 1. Elementary Examples of Integration 2. Basic Methods of Integration 3. Basic Classes of Integrable FunctionsVII. Methods of Evaluating Definite Integrals. Improper Integrals 1. Methods of Exact Evaluation of Integrals 2. Approximation Methods 3. Improper IntegralsVIII. Applications of the Integral 1. Some Problems of Geometry and Statics 2. Some Problems of PhysicsIX. Series 1. Numerical Series 2. Functional Series 3. Power Series 4. Some Applications of Taylor's Series 5. Numerical ProblemsX. Functions of Several Variables. Differential Calculus 1. Functions of Several Variables 2. Elementary Investigation of a Function 3. Derivatives and Differentials of Functions of Several Variables 4. Differentiation of Functions 5. Repeated DifferentiationXI. Applications of the Differential Calculus for Functions of Several Variables 1. Taylor's Formula. Extrema of Functions of Several Variables 2. Plane Curves 3. Vector Functions of a Scalar Argument. Curves in Space. Surfaces 4. Scalar Field. Gradient. Directional DerivativeXII. Multiple Integrals and Iterated Integration 1. Double and Triple Integrals 2. Iterated Integration 3. Integrals in Polar, Cylindrical and Spherical Coordinates 4. Applications of Double and Triple Integrals 5. Improper Integrals. Integrals Depending on a ParameterXIII. Line and Surface Integrals 1. Line Integrals 2. Coordinate Line Integrals 3. Surface IntegralsXIV. Differential equations 1. Equations of the First Order 2. Equations of the First Order (Continued) 3. Equations of the Second and Higher Orders 4. Linear Equations 5. Systems of Differential Equations 6. Numerical ProblemsXV. Trigonometric Series 1. Trigonometric Polynomials 2. Fourier Series 3. Krylov's Method. Harmonic AnalysisXVI. Elements of the Theory of FieldsAnswers Chapter I Chapter II Chapter III Chapter IV Chapter V Chapter VI Chapter VII Chapter VIII Chapter IX Chapter X Chapter XI Chapter XII Chapter XIII Chapter XIV Chapter XV Chapter XVIAppendix. Tables 1. Trigonometric Functions 2. Hyperbolic Functions 3. Reciprocals, Square and Cube Roots, Logarithms, Exponential FunctionsIndexOther Volumes in the Series in Pure and Applied Mathematics
