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Single variable calculus is typically (and reasonably) taught over a whole year, with the first semester being devoted to "differential" calculus, and the second semester being devoted to "integral" calculus.

In my own experience, "multivariable" calculus is taught in one semester. That is a course with vector calculus, partial derivatives, gradient and the chain rule. Then onto line integrals, multiple integrals, and Green's Gauss' and Stokes' theorems. That's a lot for one semester.

Would it make sense to break "multivariable up into a two semester course, differential and integral, as with single variable calculus?

That would be a "differential" course with vector calculus, partial derivatives, gradient and the chain rule, Taylor series, constrained optimization, matrices, mappings, and determinants, plus the Jacobian in the first semester. And an "integral" course with improper integration, power series, change of variables, line integrals, multiple integrals, Fourier integrals and Green's, Gauss' and Stokes' theorems in the second semester.

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The problem here is that your "differential" semester would seem to have much overlap with a linear algebra course, and removing the overlapping material would likely not leave you with enough material for an entire semester. This is one issue which doesn't arise in a quarter system: you take one quarter for differentiation and one for integration, which combined gives 20 weeks for "multivariable calculus". –  Santiago Canez Jul 14 at 17:01
    
@SantiagoCanez: Here's what happened to me. 1) differential calculus 2) integral calculus 3) "multivariate" calculus (differential and integral) 4) differential equations/linear algebra 5) "multivariate" (integral) calculus. What happened was that I got Green's and Stokes' theorems twice (3 and 5) and no introduction to constrained optimization or Jacobian determinants (should have been 3). And my "foundation" in improper integrals and power series was rather weak for 5). So why not make it "official," that is what's de facto, de jure.? –  Tom Au Jul 14 at 17:15
    
One or two semesters for which students? –  Matt F. Jul 14 at 21:19
    
Note that for many schools on the quarter system it is taught in 2 quarters, just like calc one and two are. This problem appears to be unique to the semester system. –  MHH Jul 15 at 6:04

2 Answers 2

I believe it is true that teaching the material in more time would be better. However, there are a lot of useful things for students to do with their time.

In particular, there will be a lot of pressure from your Engineering faculty against your idea. They might propose the opposite of your plan! -- more calculus material squeezed into fewer credit hours. For example, at the University of Illinois, the standard calculus sequence is:

  • 5 credits: Math 220 (Differential Calculus and many Integrals and Applications)
  • 3 credits: Math 231 (Integration Techniques, Sequences & Series)
  • 4 credits: Math 241 (Multivariable as you described)

However, this 15-credit sequence was becoming too burdensome to fit into Engineering degrees, so a special accelerated intro calculus course was created to make the engineering calculus sequence only 14 credits.

They are not evil!! and their intentions are extremely good -- the engineering people have a lot of things that they want their students to take, and it seems reasonable to them to compress first-semester calculus a bit in order to make room for another credit hour of other valuable things. Many of their students have AP credit anyway, and many of their students are fully able to succeed in the accelerated intro course.

Personally I agree with you, that something like 18 credits for all this material might be better than 15. However, the math instructor's entirely legitimate desire to teach more credits of math will come into direct conflict with everyone else's entirely legitimate desire to teach more credits of everything else. This will be the primary thing that will stop you from expanding multivariable calculus into more semesters.

One possibility might be to teach a two-semester sequence that combines multivariable calculus and the generally-boring linear algebra class into a nice soup. This kind of approach might avoid much of the counterpush by other interests.

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"Where I come from," all the courses are three credits, not four or five. So a four semester sequence would "only" be 12 credits. Maybe "single variable" goes for five credits (first semester freshman) and "multivariable" goes for five credits (second semester freshman) leaving 10 credits (instead of nine). –  Tom Au Jul 14 at 16:56
    
It might be important to distinguish between semester or quarter units since the units are not equivalent - when I've been at a school in semester units courses have been 3 units while the same class in the quarter system is usually more like 5 units. –  James S. Jul 15 at 16:25

At my University for engineering Calc I is two or three weeks of differential calculus while Calc II is multivariable and then we have a differential equations course. Not all engineers are required to take differential equations but most (possibly all) engineering majors have to take linear algebra. It used to be that Calc I was the weed out course now Calc II is. Either way adding more semesters to this sequence would lead to problems with prerequisites for major related classes that require these maths. So I guess my answer is that in the context of an engineering school slowing down this sequence will lead to a domino effect slowing down major matriculation. Anyone taking multi variable calculus should be taking it with an eye toward it being specifically useful for some other course or further math study and as a result it is a probably a class taken only by those with an aptitude for math.

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