# Calculus Volume 3

Gilbert Strang, Massachusetts Institute of Technology

Copyright Year: 2016

ISBN 13: 9781938168079

Publisher: OpenStax

Language: English

## Formats Available

## Conditions of Use

Attribution-NonCommercial-ShareAlike

CC BY-NC-SA

## Reviews

The text is very comprehensive. All of the important topics are covered and the examples are very thorough. That said, I would have liked to see something on the higher-order derivative test. The reason for the '4' in this category is that... read more

The text is very comprehensive. All of the important topics are covered and the examples are very thorough. That said, I would have liked to see something on the higher-order derivative test.

The reason for the '4' in this category is that there could be many more figures, especially in the later sections on minimizing and maximizing functions and multivariable functions (viewing the level curves in conjunction with the surface). Additionally, some figures are not entirely accurate and are misleading, e.g., the figure on continuity (the image of the open disk should not then again be an open disk on the surface, rather the image of such).

The text is very accurate.

The content is up to date. The mathematics at this level is on a firm footing and will not be changing for the foreseeable future. That said, if what one considers 'calculus 3' changes as a result of students being less prepared, then this text will need to be updated to account for students with weaker foundations. What I appreciated was the book beginning with 'parametric equations and polar coordinates.' Of course, this is suppose to be standard material in a Calculus II course, but perhaps this is evidence of "Calculus 3"-creep into "Calculus 2". I find that students are weak in this area (parametric equations) and the review would be helpful.

There are times when incorrect references are made and instructions are unclear. For example, in the minimizing/maximizing section, the 'partial derivatives test' is referenced, but never defined. Such questions in the HW will leave the students wondering what exactly is being asked. Moreover, the text makes reference to Fermat's Theorem, when they in fact meant the 2nd derivative test.

Overall, the text is fairly clear, but I think it could be edited for precision in links and wording.

There are times when the authors may become a bit too fast and loose with their terminology. Such casual use of terms is okay in conversation with the student, but is best avoided in a text.

Not fairly modular, since each section will make reference to material developed in the text. While I did not notice explicit references to Volumes 1 and 2, I would imagine those are there, as well, since it helps to make reference to earlier material developed in Calculus I and II to explain some of the concepts in Calculus III.

But, as a text for Calculus III, it is as self-contained as it can be without becoming a text for all three courses, Calculus I, II and III.

The text is very well organized and some of the application examples come at exactly the right time and at a level that is accessible to most students.

There are some issues with links to entries in the text, e.g., references to theorems are not correct. In many places, the words are poorly spaced on the lines. For example, page 710, 2nd column, third line: there are three words, "of" "semicircular" and "arch" and much whitespace in between.

Great grammar, even by non-mathematician standards! Very clear and concise wording.

Nothing struck me as being culturally insensitive, but nothing struck me as culturally inclusive, either. Really, for those that I read, the examples seem fairly neutral, as far as race/ethnicity are concerned, and focus more on the example, rather than the person behind it. For example, one of their examples mentions a company producing golf balls, not a person at the company producing golf balls. On the contrary, there are three uses of the word 'she' in the text, one use of 'woman', one use of 'man' and 27 uses of the word 'he' (32 in total, minus 5 that are in reference to the authors and donors/supporters). I would like to see more gender balance, or completely gender neutral examples.

I think this book would benefit from a thorough and complete read-through by a copy editor who is also teaching the course. Those two professions may not overlap, so I would recommend someone actually use this in teaching and detail the minor and major issues so they can be fixed in later versions.

The first 6 chapters (of 7) cover what I currently cover in a Multivariable Calculus course. Well, actually a good deal more than I cover! I would go lightly on the conic sections material and several of the sections on physics applications... read more

The first 6 chapters (of 7) cover what I currently cover in a Multivariable Calculus course. Well, actually a good deal more than I cover! I would go lightly on the conic sections material and several of the sections on physics applications (e.g. calculating moments of inertia) in order to fit things into a semester.

For this physicist, I was happy to see somewhat more mathematical rigor available than what I usually present to my students.

The 7th chapter is on differential equations.

The index on the PDF version is thoroughly hyperlinked—a joy to use.

I did not find any errors in my perusal.

I’ve found the Harvard Consortium approach useful for thinking about teaching Calculus: using symbolic / numeric / graphic / verbal modes of approaching the material. I would say this book leans most heavily on the symbolic and to a lesser extent graphic and numeric modes. It does take into account visualization technology explicitly with some exercises designated to be done with tech.

The prose is clear and definitions and theorems are carefully and precisely worded.

Despite the great number of figures and illustrations throughout the whole text, I felt like there were some sections which could have used still more. I was puzzled that in section 3.2, where the derivative and the integral of vector valued functions, and the tangent vector are all introduced, there is only one diagram.

Yes, highly consistent.

I have one quibble: Up through the first five chapters there is no typographic distinction between vectors and unit vectors. But In the section on vector calculus (Chapter 6) a caret over a vector is introduced to indicate a unit vector (in the radial direction). I would love to see this notation with the caret used throughout the text (even including the Cartesian unit vectors i, j, and k).

I thought about re-arranging things to group in another order--such as waiting to cover integration in polar coordinates (occurs very early in the book) until building up a more general expression for the differential of area--and it seems that would be need some re-working.

But, to be fair, since mathematics builds one concept on top of another, I can't imagine any textbook that can be effortlessly modular!

Overall, a solid structure and organization.

Good within current tech limits. Happy to see a lot of hyperlinks within the document and external links. Some of the simulations link to Mathematica files - proprietary but sometimes available via the CDF player. (I would love to see open source visualizations in, say, GeoGebra instead...)

(On a Mac) The PDF version, in Mac Preview, was superior to the online version

No grammar issues.

There is very little in the way of cultural reference.

The book has an absolutely lavish number of exercises (with answers to many in the back), checkpoint exercises, and examples

## Table of Contents

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 IntegralsTable of DerivativesReview of Pre-Calculus

## Ancillary Material

## 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

**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.