# Resources for teaching Riemann integration in higher dimensions and on submanifolds, with view toward Stokes' theorem?

### Question

I am looking for suggestions of good resources (textbooks or lecture notes preferably) for teaching Riemann integration in $\mathbb{R}^d$ with $d\geq 2$ and also for Riemann integration along (smooth) hypersurfaces of $\mathbb{R}^d$ (actually, all we need is Riemann integration along graphical hypersurfaces)). The goal is to have covered enough material for the students to be able to make sense of Stokes' Theorem.

### Target Audience

The target audience are students who have just finished the second semester of a sequence of classes on analysis. In the first semester they should have learned integral and differential calculus in one variable. By the end of the second semester they should have covered ordinary differential equations and differential calculus in multiple variables (rigorously with $\delta$-$\epsilon$ and such).

### Remarks

While I am fairly confident that I can write down by myself a consistent set of definitions and whatnots for Riemann integration in higher dimensions, I have not personally taken a class covering this topic (integration theory in higher dimensions is done using the Lebesgue version when I was an undergraduate), and so do not have any handy references to make sanity checks against when developing my own lecture notes. Talking to my colleagues, it seems that with the exception of the Germans my experience is rather common place. (And for reasons of personal incompetence, German lecture notes${}^\dagger$ are rather less useful to me.)

${}^\dagger$ Such as Heuser Lehrbuch der Analysis Volume 2 (Teubner), which would probably be what I'll try to parse if I can't find an English language reference.

• Have you looked at Spivak's "Calculus on Manifolds"? – kan Mar 17 '14 at 12:46
• PS. If you have specific German references, it might help some (me for example!) if you do add some pointers to them! – kan Mar 17 '14 at 12:56
• @kan: good suggestions. I'll visit the library tomorrow to take a look. – Willie Wong Mar 17 '14 at 13:48
• I don't know how closely related this is to what you have in mind, but Applied differential geometry by Burke has an interesting, unusual presentation some of this kind of material. – Ben Crowell May 15 '14 at 23:19

I notice Munkres has already been mentioned, but, I include it again:

1. Munkres is quite in depth, as I recall there are almost 100 pages devoted to the study of integration and his discussion of $n$-volumes is nice.
2. Edward's Advanced Calculus (available as Dover reprint) also has over 100 pages devoted to multivariate integration. It is not overly abstract as the manifolds considered therein are all either parameterized or level-sets of $\mathbb{R}^n$. There is a proof of the generalized Stokes Theorem in that text.
3. Hubbard and Hubbard's text is probably worth a look
4. Advanced Calculus by James J. Callahan looks to be a fairly careful and thorough treatment of low-dimensional cases.
5. Susan Colley's Vector Calculus text has a very accessible introduction to the calculus of differential forms. There are not too many calculus III texts which actually show how to integrate a $k$-form over a $k$-dimensional subspace, this one does. It is more or less at the level of an honors version of third semester calculus.

All in all, I doubt you'll be happy with any of these. I always find something is missing, or something is included I'd rather hide. So, perhaps you can write some notes on this for all of us to enjoy.

For multidimensional real analysis I recommend the two-volume bible by Duistermaat:

Features: many examples from interconnected areas, concrete computations, geometric motivation, prefers to use linear algebra in proofs. Exercises vary from very simple (to acquire familiarity with notation and language) to pretty hard (it walks you through the Peano curve construction).

With your input, these two books have everything to guide your students.

• This one slipped my mind! +1 for referring to this one! For one, there is an entertaining collection of exercises here! – kan Mar 17 '14 at 21:05
• Ah, it looks pretty nice. The only problem is that it may be a bit too high powered for the students compared to their current course. I will need to spend some time adapting the material. (Still, +1 since it would be a great reference for me.) – Willie Wong Mar 18 '14 at 8:53
• @WillieWong I suspect that is the case a.e. but I feel it is worth the effort. I already took multidimensional real analysis (not from it) and I still find it high-powered. However, I consider the ratio gain/effort to be far greater than one. Perhaps once they get used to it (with your guidance) it proves very useful. Kan mentioned Spivak: I disagree. It is a classic and as most I've encountered I feel it serves better as history. Munkres's Analysis on Manifolds is a better choice within Spivak's line of thought. – Mark Fantini Mar 18 '14 at 10:30

While it's notorious for being dense and having a lot of `slick' definitions and proofs, I really enjoyed Walter Rudin's Principles of Mathematical Analysis for its version of Riemann integration in higher dimensions (found on p. 245) and its presentation of differential forms. In fact, I couldn't understand the material in my second-semester analysis course (we used a Dover thrift edition book), and so I had to learn it on my own from this book.

I highly recommend it, but I would recommend giving many examples in class to complement the books concise presentation.

• It is in baby Rudin? /goes and flips the pages and checks/ It is in baby Rudin! How did I not notice this before? I would most likely not give the book itself to the students: Rudin is somewhat notorious for being nearly impossible to understand unless one has started from the first page and worked one's way up (especially notation-wise). But a quick glance suggests that it should be not too hard to adapt the material there for my short course, at least in terms of the fundamentals the students already know. – Willie Wong Mar 18 '14 at 8:48
• As a geometer, I find that chapter of Rudin sorely lacking. Munkres's rewrite of Spivak is more accessible. I also have written a text I use with Honors first- and second-year students. It may be a bit lower-level than what you want. – Ted Shifrin May 15 '14 at 4:10