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I was very unsatisfied with how I taught Calc III a couple years ago, and this summer I have to do it again. Are there any general resources for teaching this course? It seems like there should be, since it's a very standard thing.

In particular I'm interested in tools to show students visual graphics so they can really see what's happening in spaces we work in.

But I'm also interested in general insight on how to reach students at this level. Any tips or tricks are welcome. Anything you can point me to.

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    $\begingroup$ Welcome to ME.SE. The question is quite broad as is. Could you tell a bit more about the contents of the course (such terminology is not standard at all; maybe it is within a given country or university) and about what went wrong, so people would know what to help you with? $\endgroup$ – Tommi Brander May 29 at 9:59
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    $\begingroup$ "Are there any general resources for teaching this course?" Do you mean things like textbooks? $\endgroup$ – Joel Reyes Noche May 29 at 10:31
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    $\begingroup$ monroecc.edu/faculty/paulseeburger/calcnsf/CalcPlot3D $\endgroup$ – Steven Gubkin May 29 at 11:13
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    $\begingroup$ What does Calc III mean at your school? Multivariable calculus? Analysis? $\endgroup$ – Ben Crowell May 30 at 1:06
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    $\begingroup$ @StevenGubkin Have you used that before in a class? If so, post an answer! :) $\endgroup$ – Chris Cunningham May 30 at 12:45
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Since no one's posted an answer I will get things started with some general advice. Calculus 3 is my favorite course to teach, but it can be a bear to wrestle with the first few times. Some more specific information about your situation would be helpful (are you at Community College? Ivy League? Are the students Engineers? Math Majors? Is the class 20 students? 200?) but here is some general advice I can give you about the lecture portion of the course.

General Advice:

  • This is a course about the interplay between geometry and analysis, you need to focus on both.
  • Assign and grade homework. There is so much content students must do it progressively to keep up.
  • There are some good open source books out right now (APEX for example) but if you’re struggling to teach the course just use Stewart, it has good examples and proofs throughout and by far the most interesting selection of problems.
  • You need to do examples, you need to do them carefully, but you won't have time to do more than 1 or 2 per class, pick them wisely and don’t be afraid to use the examples in the book.
  • You can teach a semester long course without getting to Stokes Theorem, but if you need to get to Stokes Theorem, cut at the end, not the beginning. Your students have to know what a vector space is and how to perform differentiation and integration.
  • Lagrange multipliers are going to confuse the hell out of them because they don’t know multivariable algebra. Do a lot of problems connecting the math to the geometry, but also remind them of the pitfalls (IE, if you divide an equation by $y$, $y=0$ is also a solution). Give them a road map as much as possible.
  • Display your reasoning, don’t be afraid to make mistakes and fix them live. The challenge in Calc III is understanding a problem well enough to know what kind of math solves it, not plugging numbers into a formula.
  • Always leave time for questions. If you’re not getting them in Calc III you’re (almost certainly) going too fast.

Visualization:

  • Computer: Use Geogebra, MATLAB, Mathematica or whatever you're comfortable with.
  • Board: You need to be able to draw a minimum, a maximum, one general space, a saddle and a sphere. Practice those and you’ll be fine, they’ll be your work horses. Always add your axes last when you’re freehanding.
  • Add animations if you can, and some visualizations that show the interplay between the numbers, formulas and the geometry. If you're not a strong coder check out https://savkar.math.uconn.edu/calculus-3-visuals/

Other Resources:

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    $\begingroup$ I really wish more instructors would not neglect Stokes' Theorem and that general part of the course. A little less detail on multivariate limits and some of the early material can make room for it with good planning. All too often I hear stories about the poor planning of various instructors who did not do justice to Stokes' Theorem (and more generally vector calculus) in this course. As a matter of discipline and allowing for time for questions about Stokes' we ought to cover it a few days before the end at least. It can be done and done well. $\endgroup$ – James S. Cook May 31 at 15:03
  • $\begingroup$ @James S. Cook That's a good point, especially about multivariable limits, show why they're hard and move on. I should have been more specific: I would go lighter on line integrals and develop surface integrals. Line integrals can stretch out into weeks while they are conceptually very similar to surface integrals, and the related versions of Stokes. I feel like Stokes theorem is where some students really get math for the first time. $\endgroup$ – Nate Bade May 31 at 15:43
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    $\begingroup$ I think where time goes depends a lot on the taste and talents of the instructor as well as the student audience, but, it's also good to have an approximate schedule to push the course along. It is really easy to get bogged down in the multivariate integration if you get talked into teaching them integration they've forgotten. All of this advice aside, enthusiasm for the topic is worth more than any of it. $\endgroup$ – James S. Cook May 31 at 15:52
  • $\begingroup$ Just curious, what do you typically cover in a one semester calc III? I find mine ambitious but I don't touch stokes or surface integral. $\endgroup$ – Jean-Sébastien May 31 at 16:02
  • $\begingroup$ It depends on the program a bit, on 15 week semesters where students take 3 courses per semester I've done all of Part III of Stewart but it was breakneck and without daily graded homework I'm not sure students would have made it. On semester with students taking 5 courses I've done through volume integrals without touching on the line/surface integrals material. Both of these worked, although in the former students have to be told to buckle in. $\endgroup$ – Nate Bade May 31 at 18:23

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