Advanced Calculus An Introduction to Classical Analysis

ISBN-10: 0486445488
ISBN-13: 9780486445489
Edition: 2006
Authors: Louis Brand
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Description: A course in analysis dealing essentially with functions of a real variable, this text for upper-level undergraduate students introduces the basic concepts in their simplest setting and proceeds with numerous examples, theorems stated in a practical  More...

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Book details

List price: $29.95
Copyright year: 2006
Publisher: Dover Publications, Incorporated
Publication date: 1/30/2006
Binding: Paperback
Pages: 574
Size: 5.25" wide x 8.25" long x 1.25" tall
Weight: 1.562
Language: English

A course in analysis dealing essentially with functions of a real variable, this text for upper-level undergraduate students introduces the basic concepts in their simplest setting and proceeds with numerous examples, theorems stated in a practical manner, and coherently expressed proofs. 1955 edition.

The Number System
Groups
Fields
Integers
Rational Numbers
Properties of Rational Numbers
Real Numbers
Operations with Real Numbers
Complex Numbers
Complex Numbers as Vectors
Development of the Number System
Sequences and Series
Linear Point Sets
Cluster Points
Bounded Sets
Sequences
Limit of a Sequence
Operations with Limits
Fundamental Convergence Criterion
Monotone Sequences
The Number e
Nested Intervals
Infinite Series
Geometric Series
Positive Series
Telescopic Series
Comparison Tests
Cauchy's Condensation Test
Limit Form of Comparison Test
Cauchy's Root Test
d'Alembert's Ratio Test
Kummer-Jensen Tests
Gauss' Test
Absolute Convergence
Conditional Convergence
Alternating Series
Addition of Series
Rearrangement of Series
Multiplication of Series
Power Series
Binomial Series
Complex Sequences
Complex Series
Sequences and Series
Functions of a Real Variable
Functions
Limit of a Continuous Variable
Continuity at a Point
Continuity in an Interval
Bounds of a Continuous Function
Intermediate Values
Inverse Functions
Derivative
Increasing and Decreasing Functions
The Chain Rule
Derivative of an Inverse Function
Higher Derivatives
Rolle's Theorem
Theorem of Darboux
Mean-Value Theorem
The Iterative Solution of Equations
Second Mean-Value Theorem (Cauchy)
l'Hospital's Rule
The Form [infinity]/[infinity]
Other Indeterminate Forms
Approximating Polynomial
The Remainder
Taylor's Theorem
Exponential Series
Sine and Cosine Series
Even and Odd Functions
Logarithmic Series
Binomial Series
Extremes of f(x)
Summary: Functions of a Real Variable
Functions of Several Variables
Functions of Two Variables
Continuity at a Point
Continuity in a Region
Partial Derivatives
Total Differential
Differentiable Functions
Composite Functions
Functions of Three Variables
Homogeneous Functions
Higher Derivatives
Implicit Functions
One Equation
Two Equations
Three Equations
Coordinate Transformations
Jacobians
Functional Dependence
Mean-Value Theorem for f(x, y)
Taylor's Theorem for f(x, y)
Extremes of f(x, y)
Constrained Extremes
Lagrangian Multipliers
Summary: Functions of Several Variables
Vectors
Vectors
Scalar Product
Vector Product
Box Product
Derivative of a Vector
Curves
Unit Tangent Vector
Frenet's Formulas
Curvature and Torsion
Surfaces
Directional Derivative
Gradient of a Vector
Invariants of a Dyadic
Divergence and Rotation
Reciprocal Bases
Curvilinear Coordinates
Vector Algebra and Calculus
The Definite Integral
The Riemann Integral
Condition for Integrability
Integrable Functions
Integration by Summation
Properties of the Integral
Formation of Integrable Functions
The Integral as a Function of Its Upper Limit
The Fundamental Theorem of the Integral Calculus
Mean Value Theorems for Integrals
Integration by Parts
Change of Variable
Algebraic Integrals
Duhamel's Theorem
Areas
Functions of Bounded Variation
Length of a Curve
Smooth Curves
Approximate Integration
Error in Simpson's Rule
The Integral as a Function of a Parameter
Differentiation of Integrals
Application to Differential Equations
Repeated Integrals
Integrals
Improper Integrals
Types of Improper Integrals
Evaluation of Improper Integrals
Analogy with Series
Comparison Tests: Type I
Absolute Convergence: Type I
Limit Tests: Type I
Comparison Tests: Type II
Limit Tests: Type II
Conditional Convergence: Type I
Conditional Convergence: Type II
Combinations of Types I and II
Laplace Transform
Convergence of Improper Integrals
Line Integrals
Line Integrals
Line Integrals Independent of Path
Field of Force
Irrotational Vectors
Area of a Sector
Multiple Integrals
Double Integral over a Rectangle
Condition for Integrability
Continuity of an Integral
Double Integral within a Curve
Double and Repeated Integrals
Green's Theorem in the Plane
Element of Area
Change of Variables in a Double Integral
Curves on a Surface
Area of a Surface
Surface Integral
Stokes' Theorem
Line Integrals in Space
Triple Integral over a Rectangular Prism
Element of Volume
Triple and Repeated Integrals
Divergence Theorem
Solenoidal Vectors
Line, Surface, and Volume Integrals
Uniform Convergence
Reversal of Order in Limiting Processes
Uniform Convergence of a Sequence
Continuity of the Limit Function
Integrals in a Sequence
Derivatives of a Sequence
Uniform Convergence of a Series
Continuity of the Sum
Integration of Series
Differentiation of Series
Power Series
Abel's Theorem
Consequences of Abel's Theorem
Uniform Convergence of Improper Integrals
M-Test for Integrals
Continuity of Improper Integrals
Integration of Improper Integrals
Differentiation of Improper Integrals
Gamma Function
Beta Function
Relation between Beta and Gamma Function
Uniform Convergence
Functions of a Complex Variable
Rational Functions of a Complex Variable
Functions of a Complex Variable
Analytic Functions
Exponential Functions
Sine and Cosine
Hyperbolic Functions
Trigonometric Relations
Logarithm
Conformal Mapping
Definite Integrals
Cauchy's Integral Theorem
Cauchy's Integral Formula
Complex Taylor Series
Cauchy's Inequality
Isolated Singularities
Laurent Series
Bernoulli Numbers
Reciprocal of a Function
Residues
Residue Theorem
Evaluation of Definite Integrals
Improper Real Integrals
Indented Contours
Properties of Analytic Functions
Fourier Series
Orthogonal Sets of Functions
Closed and Complete Orthonormal Sets
Fourier Series
Convergence Theorem
Convergence at Discontinuities
Resolution of cot [pi]x into Partial Fractions
Approximation Theorems
Parseval's Theorem
Integration of Fourier Series
Uniform Convergence of Fourier Series
Gibbs' Phenomenon
Properties of Fourier Series
Cluster Points
Difference Equations
The Difference Calculus
Dimensional Checks
Comprehensive Test
Answers to Problems
Index

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