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A colleague of mine in a math department at another university is looking for a textbook on multivariable calculus that discusses applications of higher-dimensional integrals that feel contemporary rather than solidly traditional. In particular, he is looking for a path through the subject that culminates in something other than Stokes' theorem, since that is hard to get majors outside of math and physics excited about. The students who take the course in its current form are willing to work hard to pass the class, but they are not math-oriented (e.g., they major in economics or biology) and they simply don't see an overlap between the standard big integral theorems of vector calculus and their own interests. (I once asked a statistics professor if he ever used Stokes' theorem and he said no.) It's not even necessary that the multivariable calculus be applied to a student's major, but at least to something that looks fresh and modern (computer graphics, forecasting of all kinds) and maybe even exhibits an awareness that the people who use it and don't live in a math department rely on computers.

[Edit: My colleague is looking for a new book as part of faculty discussions to change the syllabus of the department's multivariable calculus course.]

Are there any textbooks on the market that have a genuinely different approach to what multivariable calculus can be good for and present a series of interesting applications of multidimensional integrals?

quid
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KCd
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    Well, perhaps he ought to teach variational calculus or differential equations? To reach interesting applications in biology or economics I would think differential equations is a better setting. Why bother talking about flux at all? The point of covering Stokes' and Gauss' Theorem is not an add-on to the vector calculus course. It is the course. These theorems bring clarity to what divergence and curl mean geometrically. – James S. Cook Nov 09 '14 at 06:54
  • It is not an issue of teaching a different course, but rather updating the syllabus for the multivariable calculus course. I made an edit to clarify this point. They want to include applications that will be more compelling to the students they usually have in that class (not math and physics students), since the divergence theorem and Stokes' theorem just don't mean much to the typical audience they get. – KCd Nov 09 '14 at 07:30
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    It's perhaps a step too far, but the study of multivariable/vector calculus can lead to image processing and topological data analysis, which certainly have applications in biology and many other fields outside of physics. See, for instance, Discrete Calculus by Grady & Polimeni, or Saveliev's Intelligent Perception website. – J W Nov 09 '14 at 09:58
  • Do you only want multidimensional integrals of top level forms, or do you want line integrals, surface integrals, etc? – Steven Gubkin Nov 09 '14 at 20:44
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    @StevenGubkin: Integration on hypersurfaces or along paths are both fine, but there should be contexts that don't just feed back into Green/Gauss/Stokes. There's a whole wide world of applications of integration far from physics (even in open domains of ${\mathbf R}^n$, where one doesn't need extensive "foundations" on surface elements and the like), so the request is for textbooks giving these other routes into higher-dimensional integration that will appeal to students who won't be majoring math or physics (or engineering physics). – KCd Nov 09 '14 at 23:50
  • @KCd I'm curious did you ever find such a text? I am genuinely curious as I find myself teaching multivariate calculus in the fall. As I think about your question again, maybe probability is a good source of problems. Questions about averaging over cubes, areas, space curves. Surely these all arise in the right sort of statistical questions which must surely arise in biology, business, economics etc. where statistics is key. Incidentally, thank you so much for http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/quadraticgrad.pdf it was immensely useful to me for teaching number theory... – James S. Cook May 03 '15 at 18:40
  • @JamesS.Cook, averaging over cubes or space curves?!? What exactly do you have in mind by that? Concerning your question, I never did find such a text. You can see that nobody really gave an answer to the question here, and I didn't get one anywhere else either. – KCd May 03 '15 at 21:37
  • @KCd I mean calculation of a particular average over a given geometric object. When the objects are sampled from a class of such objects I think that is probably more interesting, but, path integrals are probably wrong for calculus III. Here's a bad example of the sort of problem I'm thinking of: http://math.stackexchange.com/q/278601/36530 surely there is some big-list of these somewhere. I'm generally bad at these :) – James S. Cook May 03 '15 at 21:50
  • It is common for social science and bilogical science problems to reflect multiple factors. Just getting used to functions with more than one independent variable is useful. – guest Oct 03 '18 at 08:52

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Perhaps not really an answer to the question, but I believe I understand where this comes from.

In my experience, such calculus courses are among the first ones, long before students have seen enough (make that "anything at all" if you want) of their future field to understand what "realistic" applications of the covered techniques are. Also, such classes are usually a mixture of majors, so a relevant example for, say, chemical engineering will be greek to civil engineers or physicists. This unfortunately leaves geometric examples, or stuff that can be understood with high-school science.

Perhaps the best strategy is to talk with the teachers of the relevant higher courses to suggest problems to discuss, offer their students some "remedial" help as appropiate, or even restructure the curricula for "just in time" teaching of mathematics.

vonbrand
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