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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?

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    $\begingroup$ 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. $\endgroup$ – James S. Cook Nov 9 '14 at 6:54
  • $\begingroup$ 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. $\endgroup$ – KCd Nov 9 '14 at 7:30
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    $\begingroup$ 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. $\endgroup$ – J W Nov 9 '14 at 9:58
  • $\begingroup$ Do you only want multidimensional integrals of top level forms, or do you want line integrals, surface integrals, etc? $\endgroup$ – Steven Gubkin Nov 9 '14 at 20:44
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    $\begingroup$ @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). $\endgroup$ – KCd Nov 9 '14 at 23:50
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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.

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