Vector Calculus, 5th edition

Susan J. Colley Oberlin College
Santiago Cañez Northwestern University

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For courses in Multivariable Calculus.

Fosters a sound conceptual grasp of vector calculus

With its readable narrative, numerous figures, strong examples and exercise sets, Vector Calculus uses the language and notation of vectors and matrices to help students begin the transition from first-year calculus to more advanced technical math. Instructors will appreciate its mathematical precision, level of rigor and full selection of topics.

The 5th Edition offers clarifications, new examples and new exercises throughout. For the first time, this book is now available as a Pearson eText that includes interactive GeoGebra applets.

Hallmark features of this title

Introduction of basic linear algebra concepts  throughout shows the connection between concepts in single- and multivariable calculus.
Over 600 diagrams and figures connect analytic work to geometry and aid visualization.
Many fully worked examples throughout clarify main ideas and techniques.
Over 1400 exercises meet student needs: from practice with the basics, to applications, to mid-level exercises, to more challenging conceptual questions. Optional CAS exercises are provided.
Chapter-ending exercises help students synthesize material from multiple sections, and true/false exercises appear at the end of each chapter.
Carefully chosen advanced topics help instructors take the discussion beyond the level of other vector calculus texts.

New and updated features of this title

New derivations of the orthogonal projection formula and the Cauchy-Schwarz inequality appear in Chapter 1 (Vectors).
A description of the geometric interpretation of second-order partial derivatives has been added to Chapter 2 (Differentiation in Several Variables).
A description of the interpretation of the Lagrange multiplier has been added to Chapter 4 (Maxima and Minima in Several Variables).
Chapter 5 (Multiple Integration) adds new terminology to describe elementary regions of integration, and more examples of setting up double and triple integrals; a new subsection on probability as an application of multiple integrals; and new miscellaneous exercises on expected value.
New examples illustrating interesting uses of Green's theorem have been added to Chapter 6 (Line Integrals).
New miscellaneous exercises have been added in Chapters 1 and 4 for readers more familiar with linear algebra.

Features of Pearson eText for the 5th Edition

For the first time, this text is available as a Pearson eText, featuring a number of interactive GeoGebra applets.

1. Vectors

1.1 Vectors in Two and Three Dimensions
1.2 More About Vectors
1.3 The Dot Product
1.4 The Cross Product
1.5 Equations for Planes; Distance Problems
1.6 Some n-dimensional Geometry
1.7 New Coordinate Systems
True/False Exercises for Chapter 1
Miscellaneous Exercises for Chapter 1

2. Differentiation in Several Variables

2.1 Functions of Several Variables; Graphing Surfaces
2.2 Limits
2.3 The Derivative
2.4 Properties; Higher-order Partial Derivatives
2.5 The Chain Rule
2.6 Directional Derivatives and the Gradient
2.7 Newton's Method (optional)
True/False Exercises for Chapter 2
Miscellaneous Exercises for Chapter 2

3. Vector-Valued Functions

3.1 Parametrized Curves and Kepler's Laws
3.2 Arclength and Differential Geometry
3.3 Vector Fields: An Introduction
3.4 Gradient, Divergence, Curl, and the Del Operator
True/False Exercises for Chapter 3
Miscellaneous Exercises for Chapter 3

4. Maxima and Minima in Several Variables

4.1 Differentials and Taylor's Theorem
4.2 Extrema of Functions
4.3 Lagrange Multipliers
4.4 Some Applications of Extrema
True/False Exercises for Chapter 4
Miscellaneous Exercises for Chapter 4

5. Multiple Integration

5.1 Introduction: Areas and Volumes
5.2 Double Integrals
5.3 Changing the Order of Integration
5.4 Triple Integrals
5.5 Change of Variables
5.6 Applications of Integration
5.7 Numerical Approximations of Multiple Integrals (optional)
True/False Exercises for Chapter 5
Miscellaneous Exercises for Chapter 5

6. Line Integrals

6.1 Scalar and Vector Line Integrals
6.2 Green's Theorem
6.3 Conservative Vector Fields
True/False Exercises for Chapter 6
Miscellaneous Exercises for Chapter 6

7. Surface Integrals and Vector Analysis

7.1 Parametrized Surfaces
7.2 Surface Integrals
7.3 Stokes's and Gauss's Theorems
7.4 Further Vector Analysis; Maxwell's Equations
True/False Exercises for Chapter 7
Miscellaneous Exercises for Chapter 7

8. Vector Analysis in Higher Dimensions

8.1 An Introduction to Differential Forms
8.2 Manifolds and Integrals of k-forms
8.3 The Generalized Stokes's Theorem
True/False Exercises for Chapter 8
Miscellaneous Exercises for Chapter 8

Suggestions for Further Reading

Answers to Selected Exercises

Index

About our authors

Susan Colley is the Andrew and Pauline Delaney Professor of Mathematics at Oberlin College. She has served as chair of the department as well as editor of The American Mathematical Monthly. She received S.B. and Ph.D. degrees in mathematics from the Massachusetts Institute of Technology prior to joining the faculty at Oberlin in 1983. Her research has primarily focused on algebraic geometry, particularly enumerative problems, multiple-point singularities, and higher-order data and contact of plane curves.

Professor Colley has published papers on algebraic geometry and commutative algebra as well as articles on other mathematical subjects. She has lectured internationally on her research and has taught a wide range of subjects in undergraduate Mathematics. Professor Colley is a member of several professional and honorary societies, including the American Mathematical Society, the Mathematical Association of America, Phi Beta Kappa and Sigma Xi.

Santiago Cañez is currently a Charles Deering McCormick Distinguished Professor of Instruction in the Department of Mathematics at Northwestern University and serves as its director of undergraduate studies. He received a B.S. degree in mathematics from the University of Arizona and a Ph.D. degree in mathematics from the University of California at Berkeley prior to joining the faculty at Northwestern in 2012.

Professor Cañez works in symplectic geometry and mathematical physics and is in particular interested in groupoids and higher-order structures. He has taught a wide range of courses and has supervised numerous student theses and research projects. He is a member of the American Mathematical Society and the Mathematical Association of America.

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