Calculus Volume 3

Calculus is designed for the typical two- or three-semester general calculus course, incorporating innovative features to enhance student learning. The book guides students through the core concepts of calculus and helps them understand how those concepts apply to their lives and the world around them. Due to the comprehensive nature of the material, we are offering the book in three volumes for flexibility and efficiency. Volume 3 covers parametric equations and polar coordinates, vectors, functions of several variables, multiple integration, and second-order differential equations.

Please note that this title is published under a CC BY-NC-SA 4.0 license, which means that you are free to use and adapt, but not for commercial purposes. Changes you make need to be shared using this license.

Senior Contributing Authors

Gilbert Strang, Massachusetts Institute of Technology
Edwin “Jed” Herman, University of Wisconsin-Stevens Point


Contributing Authors

Nicoleta Virginia Bila, Fayetteville State University
Sheri J. Boyd, Rollins College
David Smith, University of the Virgin Islands
Elaine A. Terry, Saint Joseph’s University
David Torain, Hampton University
Kirsten R. Messer, Colorado State University-Pueblo
Alfred K. Mulzet, Florida State College at Jacksonville
William Radulovich, Formerly at Florida State College at Jacksonville
Erica M. Rutter, Arizona State University
David McCune, William Jewell College
Michelle Merriweather, Bronxville High School
Joseph Lakey, New Mexico State University
Julie Levandosky, Framingham State University
Valeree Falduto, Palm Beach State College
Catherine Abbott, Keuka College
Joyati Debnath, Winona State University

Chapter 1 Parametric Equations and Polar Coordinates
• Introduction
• 1.1 Parametric Equations
• 1.2 Calculus of Parametric Curves
• 1.3 Polar Coordinates
• 1.4 Area and Arc Length in Polar Coordinates
• 1.5 Conic Sections
Chapter Review
o Key Terms
o Key Equations
o Key Concepts
o Review Exercises

Chapter 2 Vectors in Space
• Introduction
• 2.1 Vectors in the Plane
• 2.2 Vectors in Three Dimensions
• 2.3 The Dot Product
• 2.4 The Cross Product
• 2.5 Equations of Lines and Planes in Space
• 2.6 Quadric Surfaces
• 2.7 Cylindrical and Spherical Coordinates
Chapter Review
o Key Terms
o Key Equations
o Key Concepts
o Review Exercises

Chapter 3 Vector-Valued Functions
• Introduction
• 3.1 Vector-Valued Functions and Space Curves
• 3.2 Calculus of Vector-Valued Functions
• 3.3 Arc Length and Curvature
• 3.4 Motion in Space
Chapter Review
o Key Terms
o Key Equations
o Key Concepts
o Review Exercises

Chapter 4 Differentiation of Functions of Several Variables
• Introduction
• 4.1 Functions of Several Variables
• 4.2 Limits and Continuity
• 4.3 Partial Derivatives
• 4.4 Tangent Planes and Linear Approximations
• 4.5 The Chain Rule
• 4.6 Directional Derivatives and the Gradient
• 4.7 Maxima/Minima Problems
• 4.8 Lagrange Multipliers
Chapter Review
o Key Terms
o Key Equations
o Key Concepts
o Review Exercises

Chapter 5 Multiple Integration
• Introduction
• 5.1 Double Integrals over Rectangular Regions
• 5.2 Double Integrals over General Regions
• 5.3 Double Integrals in Polar Coordinates
• 5.4 Triple Integrals
• 5.5 Triple Integrals in Cylindrical and Spherical Coordinates
• 5.6 Calculating Centers of Mass and Moments of Inertia
• 5.7 Change of Variables in Multiple Integrals
Chapter Review
o Key Terms
o Key Equations
o Key Concepts
o Review Exercises

Chapter 6 Vector Calculus
• Introduction
• 6.1 Vector Fields
• 6.2 Line Integrals
• 6.3 Conservative Vector Fields
• 6.4 Green’s Theorem
• 6.5 Divergence and Curl
• 6.6 Surface Integrals
• 6.7 Stokes’ Theorem
• 6.8 The Divergence Theorem
Chapter Review
o Key Terms
o Key Equations
o Key Concepts
o Review Exercises

Chapter 7 Second-Order Differential Equations
• Introduction
• 7.1 Second-Order Linear Equations
• 7.2 Nonhomogeneous Linear Equations
• 7.3 Applications
• 7.4 Series Solutions of Differential Equations
Chapter Review
o Key Terms
o Key Equations
o Key Concepts
o Review Exercises