# Calculus Volume 3

Gilbert Strang, Massachusetts Institute of Technology

Pub Date: 2016

ISBN 13: 978-1-9381680-7-9

Publisher: OpenStax

## Conditions of Use

Attribution-NonCommercial-ShareAlike

CC BY-NC-SA

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## Table of Contents

Preface

**1. Parametric Equations and Polar Coordinates**

- 1.1. Introduction
- 1.2. Parametric Equations
- 1.3. Calculus of Parametric Curves
- 1.4. Polar Coordinates
- 1.5. Area and Arc Length in Polar Coordinates
- 1.6. Conic Sections

**2. Vectors in Space**

- 2.1. Introduction
- 2.2. Vectors in the Plane
- 2.3. Vectors in Three Dimensions
- 2.4. The Dot Product
- 2.5. The Cross Product
- 2.6. Equations of Lines and Planes in Space
- 2.7. Quadric Surfaces
- 2.8. Cylindrical and Spherical Coordinates

**3. Vector-Valued Functions**

- 3.1. Introduction
- 3.2. Vector-Valued Functions and Space Curves
- 3.3. Calculus of Vector-Valued Functions
- 3.4. Arc Length and Curvature
- 3.5. Motion in Space

**4. Differentiation of Functions of Several Variables**

- 4.1. Introduction
- 4.2. Functions of Several Variables
- 4.3. Limits and Continuity
- 4.4. Partial Derivatives
- 4.5. Tangent Planes and Linear Approximations
- 4.6. The Chain Rule
- 4.7. Directional Derivatives and the Gradient
- 4.8. Maxima/Minima Problems
- 4.9. Lagrange Multipliers

**5. Multiple Integration**

- 5.1. Introduction
- 5.2. Double Integrals over Rectangular Regions
- 5.3. Double Integrals over General Regions
- 5.4. Double Integrals in Polar Coordinates
- 5.5. Triple Integrals
- 5.6. Triple Integrals in Cylindrical and Spherical Coordinates
- 5.7. Calculating Centers of Mass and Moments of Inertia
- 5.8. Change of Variables in Multiple Integrals

**6. Vector Calculus**

- 6.1. Introduction
- 6.2. Vector Fields
- 6.3. Line Integrals
- 6.4. Conservative Vector Fields
- 6.5. Green’s Theorem
- 6.6. Divergence and Curl
- 6.7. Surface Integrals
- 6.8. Stokes’ Theorem
- 6.9. The Divergence Theorem

**7. Second-Order Differential Equations**

- 7.1. Introduction
- 7.2. Second-Order Linear Equations
- 7.3. Nonhomogeneous Linear Equations
- 7.4. Applications
- 7.5. Series Solutions of Differential Equations

Table of Integrals

Table of Derivatives

Review of Pre-Calculus

## About the Book

*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.

## About the Contributors

### Author(s)

**Gilbert Strang** was an undergraduate at MIT and a Rhodes Scholar at Balliol College, Oxford. His Ph.D. was from UCLA and since then he has taught at MIT. He has been a Sloan Fellow and a Fairchild Scholar and is a Fellow of the American Academy of Arts and Sciences. He is a Professor of Mathematics at MIT, an Honorary Fellow of Balliol College, and a member of the National Academy of Sciences.

He was the President of SIAM during 1999 and 2000, and Chair of the Joint Policy Board for Mathematics. He received the von Neumann Medal of the US Association for Computational Mechanics, and the Henrici Prize for applied analysis. The first Su Buchin Prize from the International Congress of Industrial and Applied Mathematics, and the Haimo Prize from the Mathematical Association of America, were awarded for his contributions to teaching around the world.