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Schaum's Outline of Advanced Calculus, Second Edition

by: Wrede, Robert C.

Schaum's Outline of Advanced Calculus, Second Edition

by: Wrede, Robert C.

ISBN 13:
9780071375672
ISBN 10:
0071375678
Edition: 2nd
Format: Paperback
Copyright: 02/20/2002
Publisher: McGraw-Hill

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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.

Numbers

(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

(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

(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

(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

(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

Amazon no longer offers textbook rentals. We do!

Schaum's Outline of Advanced Calculus, Second Edition

Schaum's Outline of Advanced Calculus, Second Edition

9780071375672

0071375678

Summary

Table of Contents

Supplemental Materials

Write a Review

Amazon no longer offers textbook rentals. We do!

Schaum's Outline of Advanced Calculus, Second Edition

Schaum's Outline of Advanced Calculus, Second Edition

9780071375672

0071375678

Summary

Table of Contents

Supplemental Materials

Write a Review

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