Problems in Mathematical Analysis II Continuity and Differentiation 1st edition by Kaczor – Ebook PDF Instant Download/Delivery: 0821820516, 9780821820513
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• ISBN 10:0821820516
• ISBN 13:9780821820513
• Author:Kaczor
We learn by doing. We learn mathematics by doing problems. And we learn more mathematics by doing more problems. This is the sequel to “”Problems in Mathematical Analysis I”” (Volume 4 in the “”Student Mathematical Library”” series). If you want to hone your understanding of continuous and differentiable functions, this book contains hundreds of problems to help you do so. The emphasis here is on real functions of a single variable. The topics include: continuous functions, the intermediate value property, uniform continuity, mean value theorems, Taylors formula, convex functions, sequences and series of functions. The book is mainly geared toward students studying the basic principles of analysis.However, given its selection of problems, organization, and level, it would be an ideal choice for tutorial or problem-solving seminars, particularly those geared toward the Putnam exam. It is also suitable for self-study. The presentation of the material is designed to help student comprehension, to encourage them to ask their own questions, and to start research. The collection of problems will also help teachers who wish to incorporate problems into their lectures. The problems are grouped into sections according to the methods of solution. Solutions for the problems are provided. This is the sequel to “”Problems in Mathematical Analysis I”” (Volume 4 in the “”Student Mathematical Library”” series). Also available from the AMS is “”Problems in Analysis III””.
Problems in Mathematical Analysis II Continuity and Differentiation 1st Table of contents:
Chapter I
INTRODUCTION TO ANALYSIS
Sec. 1. Functions
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 Appliations 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’Hospital-Bernoulli Rule for Evaluating Indeterminate
Forms
Chapter III
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. Integration by Parts
Sec. 4. Standard Integrals Containing a Quadratic Trinomial
Sec. 5. Integration of Rational Functions
Sec. 6. Integrating Certain Irrational Functions
Sec. 7. Integrating Trigoncrretric Functions
Sec. 8. Integration of Hyperbolic Functions
Sec. 9. Using Ingonometric and Hyperbolic Substitutions for
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 VIII.
SERIES
Sec. 1. Number Series
Sec. 2. Functional Series
Sec. 3. Taylor’s Series
Sec. 4. Fourier’s Series
Chapter IX
DIFFERENTIAL EQUATIONS
Sec. 1. Verifying Solutions. Forming Differential Equations of Families of
Curves. Initial Conditions
Sec. 2. First-Order Differential Equations
Sec. 3. First-Order Diflerential 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’s Coefficients
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