Advanced Calculus by Robert C. Wrede -- Read -- Imperial Library of Trantor

Index

Cover Page Schaum’s outlines Advanced Calculus Copyright Page Preface to the Third Edition Preface to the Second Edition Contents Chapter 1 Numbers

Sets Real Numbers Decimal Representation of Real Numbers Geometric Representation of Real Numbers Operations with Real Numbers Inequalities Absolute Value of Real Numbers Exponents and Roots Logarithms Axiomatic Foundations of the Real Number System Point Sets, Intervals Countability Neighborhoods Limit Points Bounds Bolzano-Weierstrass Theorem Algebraic and Transcendental Numbers The Complex Number System Polar Form of Complex Numbers Mathematical Induction

Chapter 2 Sequences

Definition of a Sequence Limit of a Sequence Theorems on Limits of Sequences Infinity Bounded, Monotonic Sequences Least Upper Bound and Greatest Lower Bound of a Sequence Limit Superior, Limit Inferior Nested Intervals Cauchy’s Convergence Criterion Infinite Series

Chapter 3 Functions, Limits, and Continuity

Functions Graph of a Function Bounded Functions Montonic Functions Inverse Functions, Principal Values Maxima and Minima Types of Functions Transcendental Functions Limits of Functions Right- and Left-Hand Limits Theorems on Limits Infinity Special Limits Continuity Right- and Left-Hand Continuity Continuity in an Interval Theorems on Continuity Piecewise Continuity Uniform Continuity

Chapter 4 Derivatives

The Concept and Definition of a Derivative Right- and Left-Hand Derivatives Differentiability in an Interval Piecewise Differentiability Differentials The Differentiation of Composite Functions Implicit Differentiation Rules for Differentiation Derivatives of Elementary Functions Higher-Order Derivatives Mean Value Theorems L’Hospital’s Rules Applications

Chapter 5 Integrals

Introduction of the Definite Integral Measure Zero Properties of Definite Integrals Mean Value Theorems for Integrals Connecting Integral and Differential Calculus The Fundamental Theorem of the Calculus Generalization of the Limits of Integration Change of Variable of Integration Integrals of Elementary Functions Special Methods of Integration Improper Integrals Numerical Methods for Evaluating Definite Integrals Applications Arc Length Area Volumes of Revolution

Chapter 6 Partial Derivatives

Functions of Two or More Variables Neighborhoods Regions Limits Iterated Limits Continuity Uniform Continuity Partial Derivatives Higher-Order Partial Derivatives Differentials Theorems on Differentials Differentiation of Composite Functions Euler’s Theorem on Homogeneous Functions Implicit Functions Jacobians Partial Derivatives Using Jacobians Theorems on Jacobians Transformations Curvilinear Coordinates Mean Value Theorems

Chapter 7 Vectors

Vectors Geometric Properties of Vectors Algebraic Properties of Vectors Linear Independence and Linear Dependence of a Set of Vectors Unit Vectors Rectangular (Orthogonal) Unit Vectors Components of a Vector Dot, Scalar, or Inner Product Cross or Vector Product Triple Products Axiomatic Approach To Vector Analysis Vector Functions Limits, Continuity, and Derivatives of Vector Functions Geometric Interpretation of a Vector Derivative Gradient, Divergence, and Curl Formulas Involving V Vector Interpretation of Jacobians and Orthogonal Curvilinear Coordinates Gradient, Divergence, Curl, and Laplacian in Orthogonal Curvilinear Coordinates Special Curvilinear Coordinates

Chapter 8 Applications of Partial Derivatives

Applications To Geometry Directional Derivatives Differentiation Under the Integral Sign Integration Under the Integral Sign Maxima and Minima Method of Lagrange Multipliers for Maxima and Minima Applications To Errors

Chapter 9 Multiple Integrals

Double Integrals Iterated Integrals Triple Integrals Transformations of Multiple Integrals The Differential Element of Area in Polar Coordinates, Differential Elements of Area in Cylindrical and Spherical Coordinates

Chapter 10 Line Integrals, Surface Integrals, and Integral Theorems

Line Integrals Evaluation of Line Integrals for Plane Curves Properties of Line Integrals Expressed for Plane Curves Simple Closed Curves, Simply and Multiply Connected Regions Green’s Theorem in the Plane Conditions for a Line Integral To Be Independent of the Path Surface Integrals The Divergence Theorem Stokes’s Theorem

Chapter 11 Infinite Series

Definitions of Infinite Series and Their Convergence and Divergence Fundamental Facts Concerning Infinite Series Special Series Tests for Convergence and Divergence of Series of Constants Theorems on Absolutely Convergent Series Infinite Sequences and Series of Functions, Uniform Convergence Special Tests for Uniform Convergence of Series Theorems on Uniformly Convergent Series Power Series Theorems on Power Series Operations with Power Series Expansion of Functions in Power Series Taylor’s Theorem Some Important Power Series Special Topics Taylor’s Theorem (For Two Variables)

Chapter 12 Improper Integrals

Definition of an Improper Integral Improper Integrals of the First Kind (Unbounded Intervals) Convergence or Divergence of Improper Integrals of the First Kind Special Improper Integers of the First Kind Convergence Tests for Improper Integrals of the First Kind Improper Integrals of the Second Kind Cauchy Principal Value Special Improper Integrals of the Second Kind Convergence Tests for Improper Integrals of the Second Kind Improper Integrals of the Third Kind Improper Integrals Containing a Parameter, Uniform Convergence Special Tests for Uniform Convergence of Integrals Theorems on Uniformly Convergent Integrals Evaluation of Definite Integrals Laplace Transforms Linearity Convergence Application Improper Multiple Integrals

Chapter 13 Fourier Series

Periodic Functions Fourier Series Orthogonality Conditions for the Sine and Cosine Functions Dirichlet Conditions Odd and Even Functions Half Range Fourier Sine or Cosine Series Parseval’s Identity Differentiation and Integration of Fourier Series Complex Notation for Fourier Series Boundary-Value Problems Orthogonal Functions

Chapter 14 Fourier Integrals

The Fourier Integral Equivalent Forms of Fourier’s Integral Theorem Fourier Transforms

Chapter 15 Gamma and Beta Functions

The Gamma Function Table of Values and Graph of the Gamma Function The Beta Function Dirichlet Integrals

Chapter 16 Functions of a Complex Variable

Functions Limits and Continuity Derivatives Cauchy-Riemann Equations Integrals Cauchy’s Theorem Cauchy’s Integral Formulas Taylor’s Series Singular Points Poles Laurent’s Series Branches and Branch Points Residues Residue Theorem Evaluation of Definite Integrals

Index