Schaum’s Outline of Advanced Calculus, Third Edition
by: Robert Wrede, Murray Spiegel
Abstract: Study faster, learn better, and get top grades. Schaum’s Outline of Advanced Calculus mirrors the course in scope and sequence to help you understand basic concepts and offer extra practice on topics such as derivatives, integrals, multiple integrals, applications of partial derivatives, vectors, improper integrals, and Fourier series. Coverage will also include linear independence and linear dependence of a set of vectors, method of Lagrange multipliers for maxima and minima, the divergence theorem, and orthogonality conditions for the sine and cosine functions.
Full details
Table of Contents
- A. Preface to the Third Edition
- B. Preface to the Second Edition
- 1. Numbers
- 2. Sequences
- 3. Functions, Limits, and Continuity
- 4. Derivatives
- 5. Integrals
- 6. Partial Derivatives
- 7. Vectors
- 8. Applications of Partial Derivatives
- 9. Multiple Integrals
- 10. Line Integrals, Surface Integrals, and Integral Theorems
- 11. Infinite Series
- 12. Improper Integrals
- 13. Fourier Series
- 14. Fourier Integrals
- 15. Gamma and Beta Functions
- 16. Functions of a Complex Variable
Tools & Media
Expanded Table of Contents
- A. Preface to the Third Edition
- B. Preface to the Second Edition
- 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
- SOLVED PROBLEMS
- SUPPLEMENTARY PROBLEMS
- 2. Sequences
- 3. Functions, Limits, and Continuity
- Functions
- Graph of a Function
- Bounded Functions
- Monotonic 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
- SOLVED PROBLEMS
- SUPPLEMENTARY PROBLEMS
- 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
- SOLVED PROBLEMS
- SUPPLEMENTARY PROBLEMS
- 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
- SOLVED PROBLEMS
- SUPPLEMENTARY PROBLEMS
- 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 Theorem
- SOLVED PROBLEMS
- SUPPLEMENTARY PROBLEMS
- 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
- Formulas Involving ∇
- Vector Interpretation of Jacobians and Orthogonal Curvilinear Coordinates
- Gradient Divergence, Curl, and Laplacian in Orthogonal Curvilinear Coordinates
- Special Curvilinear Coordinates
- SOLVED PROBLEMS
- SUPPLEMENTARY PROBLEMS
- 8. Applications of Partial Derivatives
- 9. Multiple Integrals
- 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
- SOLVED PROBLEMS
- SUPPLEMENTARY PROBLEMS
- 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)
- SOLVED PROBLEMS
- Convergence And Divergence Of Series Of Constants
- 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 Integrals 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
- SOLVED PROBLEMS
- Improper integrals of the first kind
- SUPPLEMENTARY PROBLEMS
- 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
- SOLVED PROBLEMS
- SUPPLEMENTARY PROBLEMS
- 14. Fourier Integrals
- 15. Gamma and Beta Functions
- 16. Functions of a Complex Variable
Book Details
Title: Schaum’s Outline of Advanced Calculus, Third Edition
Publisher: : New York, Chicago, San Francisco, Lisbon, London, Madrid, Mexico City, Milan, New Delhi, San Juan, Seoul, Singapore, Sydney, Toronto
Copyright / Pub. Date: 2010, 2002, 1963 by the McGraw-Hill Companies, Inc
ISBN: 9780071623667
Authors:
Robert Wrede received his B.S. and M.A. degrees from Miami University, Oxford, Ohio. After teaching there for a year, he attended Indiana University and was awarded a Ph.D. in mathematics. He taught at San Jose State University from 1955 to 1994. He also consulted at IBM, the Naval Radiation Laboratory at Hunter’s Point, and with several textbook companies. His primary interests have been in tensor analysis and relativity theory.
Murray Spiegel is the author of this McGraw-Hill Professional publication.
Description: Study faster, learn better, and get top grades. Schaum’s Outline of Advanced Calculus mirrors the course in scope and sequence to help you understand basic concepts and offer extra practice on topics such as derivatives, integrals, multiple integrals, applications of partial derivatives, vectors, improper integrals, and Fourier series. Coverage will also include linear independence and linear dependence of a set of vectors, method of Lagrange multipliers for maxima and minima, the divergence theorem, and orthogonality conditions for the sine and cosine functions.
