Welcome to Calculus Volume 3, an OpenStax resource. This textbook was written to increase student access to high-quality learning materials, maintaining highest standards of academic rigor at little to no cost.
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About 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.
Coverage and scope
Our Calculus Volume 3 textbook adheres to the scope and sequence of most general calculus courses nationwide. We have worked to make calculus interesting and accessible to students while maintaining the mathematical rigor inherent in the subject. With this objective in mind, the content of the three volumes of Calculus have been developed and arranged to provide a logical progression from fundamental to more advanced concepts, building upon what students have already learned and emphasizing connections between topics and between theory and applications. The goal of each section is to enable students not just to recognize concepts, but work with them in ways that will be useful in later courses and future careers. The organization and pedagogical features were developed and vetted with feedback from mathematics educators dedicated to the project.
- Chapter 1: Functions and Graphs
- Chapter 2: Limits
- Chapter 3: Derivatives
- Chapter 4: Applications of Derivatives
- Chapter 5: Integration
- Chapter 6: Applications of Integration
- Chapter 1: Integration
- Chapter 2: Applications of Integration
- Chapter 3: Techniques of Integration
- Chapter 4: Introduction to Differential Equations
- Chapter 5: Sequences and Series
- Chapter 6: Power Series
- Chapter 7: Parametric Equations and Polar Coordinates
- Chapter 1: Parametric Equations and Polar Coordinates
- Chapter 2: Vectors in Space
- Chapter 3: Vector-Valued Functions
- Chapter 4: Differentiation of Functions of Several Variables
- Chapter 5: Multiple Integration
- Chapter 6: Vector Calculus
- Chapter 7: Second-Order Differential Equations
Throughout Calculus Volume 3 you will find examples and exercises that present classical ideas and techniques as well as modern applications and methods. Derivations and explanations are based on years of classroom experience on the part of long-time calculus professors, striving for a balance of clarity and rigor that has proven successful with their students. Motivational applications cover important topics in probability, biology, ecology, business, and economics, as well as areas of physics, chemistry, engineering, and computer science. Student Projects in each chapter give students opportunities to explore interesting sidelights in pure and applied mathematics, from navigating a banked turn to adapting a moon landing vehicle for a new mission to Mars. Chapter Opening Applications pose problems that are solved later in the chapter, using the ideas covered in that chapter. Problems include the average distance of Halley's Comment from the Sun, and the vector field of a hurricane. Definitions, Rules, and Theorems are highlighted throughout the text, including over 60 Proofs of theorems.
Assessments that reinforce key concepts
In-chapter Examples walk students through problems by posing a question, stepping out a solution, and then asking students to practice the skill with a “Checkpoint” question. The book also includes assessments at the end of each chapter so students can apply what they’ve learned through practice problems. Many exercises are marked with a [T] to indicate they are suitable for solution by technology, including calculators or Computer Algebra Systems (CAS). Answers for selected exercises are available in the Answer Key at the back of the book. The book also includes assessments at the end of each chapter so students can apply what they’ve learned through practice problems.
Early or late transcendentals
The three volumes of Calculus are designed to accommodate both Early and Late Transcendental approaches to calculus. Exponential and logarithmic functions are introduced informally in Chapter 1 of Volume 1 and presented in more rigorous terms in Chapter 6 in Volume 1 and Chapter 2 in Volume 2. Differentiation and integration of these functions is covered in Chapters 3–5 in Volume 1 and Chapter 1 in Volume 2 for instructors who want to include them with other types of functions. These discussions, however, are in separate sections that can be skipped for instructors who prefer to wait until the integral definitions are given before teaching the calculus derivations of exponentials and logarithms.
Comprehensive art program
Our art program is designed to enhance students’ understanding of concepts through clear and effective illustrations, diagrams, and photographs.
Student and instructor resources
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About the authors
Senior contributing authors
Gilbert Strang, Massachusetts Institute of Technology
Dr. Strang received his PhD from UCLA in 1959 and has been teaching mathematics at MIT ever since. His Calculus online textbook is one of eleven that he has published and is the basis from which our final product has been derived and updated for today’s student. Strang is a decorated mathematician and past Rhodes Scholar at Oxford University.
Edwin “Jed” Herman, University of Wisconsin-Stevens Point
Dr. Herman earned a BS in Mathematics from Harvey Mudd College in 1985, an MA in Mathematics from UCLA in 1987, and a PhD in Mathematics from the University of Oregon in 1997. He is currently a Professor at the University of Wisconsin-Stevens Point. He has more than 20 years of experience teaching college mathematics, is a student research mentor, is experienced in course development/design, and is also an avid board game designer and player.
Catherine Abbott, Keuka College
Nicoleta Virginia Bila, Fayetteville State University
Sheri J. Boyd, Rollins College
Joyati Debnath, Winona State University
Valeree Falduto, Palm Beach State College
Joseph Lakey, New Mexico State University
Julie Levandosky, Framingham State University
David McCune, William Jewell College
Michelle Merriweather, Bronxville High School
Kirsten R. Messer, Colorado State University - Pueblo
Alfred K. Mulzet, Florida State College at Jacksonville
William Radulovich (retired), Florida State College at Jacksonville
Erica M. Rutter, Arizona State University
David Smith, University of the Virgin Islands
Elaine A. Terry, Saint Joseph’s University
David Torain, Hampton University
Marwan A. Abu-Sawwa, Florida State College at Jacksonville
Kenneth J. Bernard, Virginia State University
John Beyers, University of Maryland
Charles Buehrle, Franklin & Marshall College
Matthew Cathey, Wofford College
Michael Cohen, Hofstra University
William DeSalazar, Broward County School System
Murray Eisenberg, University of Massachusetts Amherst
Kristyanna Erickson, Cecil College
Tiernan Fogarty, Oregon Institute of Technology
David French, Tidewater Community College
Marilyn Gloyer, Virginia Commonwealth University
Shawna Haider, Salt Lake Community College
Lance Hemlow, Raritan Valley Community College
Jerry Jared, The Blue Ridge School
Peter Jipsen, Chapman University
David Johnson, Lehigh University
M.R. Khadivi, Jackson State University
Robert J. Krueger, Concordia University
Tor A. Kwembe, Jackson State University
Jean-Marie Magnier, Springfield Technical Community College
Cheryl Chute Miller, SUNY Potsdam
Bagisa Mukherjee, Penn State University, Worthington Scranton Campus
Kasso Okoudjou, University of Maryland College Park
Peter Olszewski, Penn State Erie, The Behrend College
Steven Purtee, Valencia College
Alice Ramos, Bethel College
Doug Shaw, University of Northern Iowa
Hussain Elalaoui-Talibi, Tuskegee University
Jeffrey Taub, Maine Maritime Academy
William Thistleton, SUNY Polytechnic Institute
A. David Trubatch, Montclair State University
Carmen Wright, Jackson State University
Zhenbu Zhang, Jackson State University