Calculus Volume 3

Reviewed by Rob Niemeyer, Assistant Professor, Metropolitan State University of Denver on 10/21/19

Comprehensiveness rating: 4 see less

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).

Content Accuracy rating: 5

The text is very accurate.

Relevance/Longevity rating: 5

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.

Clarity rating: 4

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.

Consistency rating: 4

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.

Modularity rating: 2

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.

Organization/Structure/Flow rating: 5

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.

Interface rating: 4

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.

Grammatical Errors rating: 5

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

Cultural Relevance rating: 3

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.

Comments

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.

Reviewed by Paul Meyer Reimer, Professor of Physics, Goshen College on 7/15/19

Comprehensiveness rating: 5 see less

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.

Content Accuracy rating: 5

I did not find any errors in my perusal.

Relevance/Longevity rating: 4

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.

Clarity rating: 4

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.

Consistency rating: 5

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).

Modularity rating: 4

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!

Organization/Structure/Flow rating: 5

Overall, a solid structure and organization.

Interface rating: 5

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

Grammatical Errors rating: 5

No grammar issues.

Cultural Relevance rating: 4

There is very little in the way of cultural reference.

Comments

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