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Advanced Calculus

ISBN-10: 0201799375
ISBN-13: 9780201799378
Edition: 5th 2003 (Revised)
Authors: Wilfred Kaplan
List price: $117.80 Buy it from $71.74
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Book details

List price: $117.80
Edition: 5th
Copyright year: 2003
Publisher: Addison Wesley
Publication date: 7/25/2002
Binding: Paperback
Pages: 736
Size: 6.75" wide x 9.75" long x 1.50" tall
Weight: 2.772
Language: English

Vectors and Matrices
Introduction
Vectors in Space
Linear Independence
Lines and Planes
Determinants
Simultaneous Linear Equations
Matrices
Addition of Matrices
Scalar Times Matrix
Multiplication of Matrices
Inverse of a Square Matrix
Gaussian Elimination
Eigenvalues of a Square Matrix
The Transpose
Orthogonal Matrices
Analytic Geometry and Vectors n-Dimensional Space
Axioms for Vn
Linear Mappings
Subspaces
Rank of a Matrix
Other Vector Spaces
Differential Calculus of Functions of Several Variables
Functions of Several Variables
Domains and Regions
Functional Notation Level Curves and Level Surfaces
Limits and Continuity
Partial Derivatives
Total Differential Fundamental Lemma
Differential of Functions of n Variables
The Jacobian Matrix
Derivatives and Differentials of Composite Functions
The General Chain Rule
Implicit Functions
Proof of a Case of the Implicit Function Theorem
Inverse Functions
Curvilinear Coordinates
Geometrical Applications
The Directional Derivative
Partial Derivatives of Higher Order
Higher Derivatives of Composite Functions
The Laplacian in Polar, Cylindrical, and Spherical Coordinates
Higher Derivatives of Implicit Functions
Mixima and Minima of Functions of Several Variables
Extrema for Functions with Side Conditions
Lagrange Multipliers
Maxima and Minima of Quadratic Forms on the Unit Sphere
Functional Dependence
Real Variable Theory
Theorem on Maximum and Minimum
Vector Differential Calculus
Introduction
Vector Fields and Scalar Fields
The Gradient Field
The Divergence of a Vector Field
The Curl of a Vector Field
Combined Operations
Curvilinear Coordinates in Space
Orthogonal Coordinates
Vector Operations in Orthogonal Curvilinear Coordinates
Tensors
Tensors on a Surface or Hypersurface
Alternating Tensors
Exterior Product
Integral Calculus of Functions of Several Variables
The Definite Integral
Numerical Evaluation of Indefinite Integrals
Elliptic Integrals
Double Integrals
Triple Integrals and Multiple Integrals in General
Integrals of Vector Functions
Change of Variables in Integrals
Arc Length and Surface Area
Improper Multiple Integrals
Integrals depending on a Parameter Leibnitz's Rule
Uniform Continuity
Existence of the Riemann Integral
Theory of Double Integrals
Vector Integral Calculus
Two-Dimensional Theory
Introduction
Line Integrals in the Plane
Integrals with Respect to Arc Length
Basic Properties of Line Integrals
Line Integrals as Integrals of Vectors
Green's Theorem
Independence of Path
Simply Connected Domains
Extension of Results to Multiply Connected Domains
Three-Dimensional Theory and Applications
Line Integrals in Space
Surfaces in Space
Orientability
Surface Integrals
The Divergence Theorem
Stokes's Theorem
Integrals Independent of Path
Irrotational and Solenoidal Fields
Change of Variables in a Multiple Integral
Physical Applications
Potential Theory in the Plane
Green's Third Identity
Potential Theory in Space
Differential Forms
Change of Variables in an m-Form and General Stokes's Theorem
Tensor Aspects of Differential Forms
Tensors and Differential Forms without Coordinates
Infinite Series
Introduction
Infinite Sequences
Upper and Lower Limits
Further Properties of Sequences
Infinite Series
Tests for Convergence and Divergence
Examples and Applications of Tests for Convergence and Divergence
Extended Ratio Test and Root Test
Computation with Series Estimate of Error
Operations on Series
Sequences and Series of Functions
Uniform Convergence
Weierstrass M-Test for Uniform Convergence
Properties of Uniformly Convergent Series and Sequences
Power Series
Taylor and MacLaurin Series
Taylor's Formula with Remainder
Further Operations on Power Series
Sequences and Series of Complex Numbers
Sequences and Series of Functions of Several Variables
Taylor's Formula for Functions of Several Variables
Improper Integrals Versus Infinite Series
Improper Integrals Depending on a Parameter Uniform Convergence
Principal Value of Improper Integrals
Laplace Transformation
<F128>G-Function and B-Function
Convergence of Improper Multiple Integrals
Fourier Series and Orthogonal Functions
Trigonometric Series
Fourier Series
Convergence of Fourier Series
Examples Minimizing of Square Error
Generalizations
Fourier Cosine Series
Fourier Sine Series
Remarks on Applications of Fourier Series
Uniqueness Theorem
Proof of Fundamental Theorem for Continuous, Periodic, and Piecewise Very Smooth Functions
Proof of Fundamental Theorem
Orthogonal Functions
Fourier Series of Orthogonal Functions
Completeness
Sufficient Conditions for Completeness
Integration and Differentiation of Fourier Series
Fourier-Legendre Series
Fourier-Bessel Series
Orthogonal Systems of Functions of Several Variables
Complex Form of Fourier Series
Fourier Integral
The Laplace Transform as a Special Case of the Fourier Transform
General Functions
Functions of a Complex Variable
Complex Functions
Complex-Valued Functions of a Real Variable
Complex-Valued Functions of a Complex Variable
Limits and Continuity
Derivatives and Differentials
Integrals
Analytic Functions
Cauchy-Riemann Equations
The Functions log z, az, za, sin-1 z, cos-1 z
Integrals of Analytic Functions
Cauchy Integral Theorem
Cauchy's Integral Formula
Power Series as Analytic Functions
Power Series Expansion of General Analytic Function
Power Series in Positive and Negative Powers
Laurent Expansion
Isolated Singularities of an Analytic Function
Zeros and Poles
The Complex Number <F128>à
Residues
Residue at Infinity
Logarithmic Residues
Argument Principle
Partial Fraction Expansion of Rational Functions
Application of Residues to Evaluation of Real Integrals
Definition of Conformal Mapping
Examples of Conformal Mapping
Applications of Conformal Mapping
The Dirichlet Problem
Dirichlet Problem for the Half-Plane
Conformal Mapping in Hydrodynamics
Applications of Conformal Mapping in the Theory of Elasticity
Further Applications of Conformal Mapping
General Formulas for One-to-One Mapping
Schwarz-Christoffel Transformation
Ordinary Differential Equations
Differential Equations
Solutions
The Basic Problems
Existence Theorem
Linear Differential Equations
Systems of Differential Equations
Linear Systems
Linear Systems with Constant Coefficients
A Class of Vibration Problems
Solution of Differential Equations by Taylor Series
The Existence and Uniqueness Theorem
Partial Differential Equations
Introduction
Review of Equation for Forced Vibrations of a Spring
Case of Two Particles
Case of n Particles
Continuous Medium
Fundamental Partial Differential Equation
Classification of Partial Differential Equations
Basic Problems
The Wave Equation in One Dimension
Harmonic Motion
Properties of Solutions of the Wave Equation
The One-Dimensional Heat Equation
Exponential Decay
Properties of Solutions of the Heat Equation
Equilibrium and Approach to Equilibrium
Forced Motion
Equations with Variable Coefficients
Sturm-Liouville Problems
Equations in Two and Three Dimensions
Separation of Variables
Unbounded Regions
Continuous Spectrum
Numerical Methods
Variational Methods
Partial Differential Equations and Integral Equations

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