Schaum's Outline of Advanced Calculus, Second Edition

by Wrede, Robert C.

• ISBN13: 9780071375672
• ISBN10: 0071375678
• Edition: 2nd
• Format: Paperback
• Copyright: 2002-02-20
• Publisher: McGraw-Hill

Summary

This best-selling guide--which has sold more than 340,000 copies since its first publication--has been thoroughly updated throughout to correspons to current advanced calculus courses. A complete and comprehensive review of the subject, this updated edition features important new chapters on topology and Laplace transforms and essential new theorems, with explanatory proofs.

Table of Contents

Numbers	1	(22)
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
Sequences	23	(16)
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
Functions, Limits, and Continuity	39	(26)
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
Derivatives	65	(25)
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
Integrals	90	(26)
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
Partial Derivatives	116	(34)
Functions of two or more variables
Three-dimensional rectangular coordinate systems
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
Transformation
Curvilinear coordinates
Mean value theorems
Vectors	150	(33)
Vectors
Geometric properties
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 or scalar 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 ∇
Vector interpretation of Jacobians, Orthogonal curvilinear coordinates
Gradient, divergence, curl, and Laplacian in orthogonal curvilinear coordinates
Special curvilinear coordinates
Applications of Partial Derivatives	183	(24)
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
Multiple Integrals	207	(22)
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
Line Integrals, Surface Integrals, and Integral Theorems	229	(36)
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
Stoke's theorem
Infinite Series	265	(41)
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)
Improper Integrals	306	(30)
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
Fourier Series	336	(27)
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
Fourier Integrals	363	(12)
The Fourier integral
Equivalent forms of Fourier's integral theorem Fourier transforms
Gamma and Beta Functions	375	(17)
The gamma function
Table of values and graph of the gamma function
The beta function
Dirichlet integrals
Functions of a Complex Variable	392	(33)
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	425