A question about Vector Analysis problems

Why is it difficult to find really challenging vector analysis problems (problems about Green's, Stokes' and Gauss' theorems in a Calculus 3 course) in Calculus books? Most of the problems are elementary, at least that's the impression I have(I could be wrong). Is it really difficult to formulate new and interesting problems in this part(Vector Analysis), or do the authors try to go very slowly on this part of the subject?

• Maybe the same thing as with arc-length problems. Problems where the answer can be computed by hand are only very simple ones. Oct 5 '17 at 12:49
• When looking for more advanced problems on various calculus topics I usually just went to the university library (and after many years, my own library was often more than adequate) and flipped through the many advanced calculus texts on the shelves. Also, keep in mind that while the problems may seem "old and uninteresting to you", for students the problems will be new (but probably not interesting, but this is also likely to be true for what you think will be interesting). Remember that unlike you, your students haven't been seeing these same problems every semester for many years. Oct 6 '17 at 7:57
• Have you looked at thermodynamics, fluid mechanics, or electrical engineering texts for problems? These three theorems are closely related to the Reynolds Transport Theorem, which (from my engineering point of view) is one of the two most important theorems in Calculus (along with the Fundamental Theorem of Calculus). The Reynolds Transport Theorem is the basis for most quantitative problems in physics, chemistry, and engineering. Jun 7 '18 at 0:08

You are correct. There are not nearly enough interesting problems typically given for the major theorems of vector calculus. For the most part, the problems I find are not that interesting and mostly amount to "verify the theorem" or some such thing. Of course, there is more. Where to find it? I'm not sure I know a good answer to that off hand, I'm always working to find more such problems. In any event, this is symptomatic of a larger problem with the last third of mainstream calculus texts.

What commonly passes for multivariate calculus in the standard textbooks is horrible. There is far too little attention paid to the parametric approach. Hardly any of the books do non-cartesian coordinates justice. We get only 20 pages or so for the crowning jewels of Green, Stokes and Divergence. Deformation theorems are given little coverage. Earlier in the course, many books give little attention to Frenet frames and the geometry of curves. All of these are helpful to understand how we actually apply vector calculus to physics. Furthermore, what is even less likely to find, the foundations of potential theory including Green's 2nd and 3rd identities and Helmholtz Theorem. I haven't even begun to scratch the surface here, there is tons missing for differentiability as well. So, what to do?

Simple. Ignore the required text and make your own path. Refuse to accept the sad watered down fake calculus III which is pushed on us by the standard texts where calculus III is essentially an afterthought.

One refreshing counterpoint to my complaints here would be Susan Colley's excellent Vector Calculus text. It actually has the same sort of detail for calculus III as is easily found for Calculus I and II in many texts. I really wish my school split off the text for Calculus III because I am universally displeased with the standard texts.

Incidentally, my displeasure stems not initially from my mathematics. Rather, it was as a physics student it became apparent that much of the real calculational techniques I needed were (for reasons indefensible to me) simply removed from my education which was ostensibly complete.

With a little effort you could include a generalization of the directional derivative towards the (std) covariant derivative: For two vector fields $F,G:\Bbb R^3\to\Bbb R^3$ the construction $$D_FG=[JG]F$$ captures how the components of $G$ vary in the $F$ direction.

With that you could ask for example: How the normal field of a surface varies in the tangential direction giving you the Weingarten's relations (or the Gauss' map, which is the derivative of the Shape operator). From the Weingarten's relations you get a $2\times2$ matrix whose determinant defines the Gaussian Curvature of the surface.

The algebraic, differential and geometrical elementary concepts combine on a clever fashion giving a wide range of cases to be amused at the exercising activities.

• extremely important to be well trained of the chain's rule which isn't very-very-very-difficult to grasp : ) Jun 6 '18 at 20:11

I suspect calc 3 suffers a little from not having enough time to cover all the topics and really go in some depth on them. A compromise is made where students get at least an exposure (but not mastery) to Stokes and the like, towards the end of their class. I don't know a good improvement for this given the time available and the content that needs to be covered and that while the topic may be easy for math profs it is always new and hard for the endless newly birthed students. By analogy same issue exists with Laplace transforms in diffyscrews, where you just do an exposure since there is not time for more and at least that way, they've heard of it (and can learn more in an EE or ME controls class if needed).

• The answer is not too complicated. We just have to have discipline as instructors to not spend too much time on differentiation and integration of functions of several variables. We just have to make the time for Vector Calculus. We must set aside rote practice in class and relegate some of it to homework. Nov 7 '17 at 5:19

FWIW, physics students get a lot more exposure, practice, and harder problems in junior year classical E&M.

Just check out the first 40 pages of Wangsness. (And that's the review, the rest of the whole book is just practicing more vector calc.)

https://www.scribd.com/document/342675118/Roald-K-Wangsness-Electromagnetic-Fields-pdf