Vector Calculus, 5th edition

Published by Pearson (August 15, 2022) © 2022

  • Susan J. Colley Oberlin College
  • Santiago Cañez Northwestern University
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Vector Calculus uses the language and notation of vectors and matrices to help you start the transition from first-year calculus to more advanced technical math. Its readable narrative, numerous figures, strong examples and exercise sets help foster a solid grasp of the concepts. An appropriate level of rigor collects most of the technical derivations at the ends of sections; many proofs are available for reference but are positioned in a way that does not disrupt the flow of main ideas. 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. Background in single-variable calculus is assumed.

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

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