What Should be there in a Single Variable Calculus course for Engineers?

Question

Thought a little background is required.

The place I teach students learn Calculus in school, not the theory but only the computational aspect.

I am trying to teach Calculus to first year Engineering students using Apostol but there is huge resistance from my own Department. The arguments range from Engineers don't need this, they won't understand theory and its useless for them, they should focus more on concepts and visualization (geometric intuition). I am feeling a bit defenseless and cornered as I never been part of discussions where people discussed pedagogy.

All I know, I spent 15 odd years in an Engineering Institute and 95% percent of my friends are Engineers and they mostly know/did their maths courses rather well; which was a proper course, as in they did proofs, knew about the real number system (lub axiom) , Sequences (Bolzano Weierstrass) , Continuity, Roll's Theorem and MVT, Taylor's Expansion all, Fundamental Theorem of Calculus and they did all these in their first semester along with proofs.

Here I am facing a scenario where Cauchy Sequence is not to be touched on as if it is going to make students head explode. Taylor's Expansion is not to be talked about, forget Taylor's Series I am not allowed to teach Series. The problem is the resistance comes from my own head/department and not the Engineering Department.

I am totally confused and do not know how can I argue my case. Apparently all these is part of Real Analysis and not Calculus and also Proofs are not required, how does one argue against this ?

I am not complaining at all, I am asking for advise regarding how to present my case in a compelling and convincing way.

Regards Vb

Added in Response to comment by Douglas Zare

those with different preparation would probably go through a preparatory course I suppose. What I mean to what extent do the Engineering students need to know Calculus to do well in their Career. What is the basic minimum. Where they begin is a different issue, what's the least they must & should know ?

Answer 1

I strongly agree with Douglas Zare that it is highly dependent on what incoming students are prepared for in terms of their mathematical background beforehand. As far what is the minimum calculus knowledge that engineers need, this too is dependent on what field of engineering they go into. I know several engineers that use no more than basic trigonometry in what they do and do not need to know much (if any) calculus, though that will certainly not be true for all engineers.

In the US (where I teach), many schools make a distinction between a calculus course and a real analysis course. A calculus course is essentially a problem-solving course, in which students are introduced to basic facts related to the operations of differentiation and integration, with a few supplementary facts about underlying notions such as the real number line and limits, and then expected to be able to solve a wide range of problems that, in my view at least, often range from essential calculations that everyone should be able to do to esoteric trivia that has remained in the curriculum due to inertia. On the other hand, a real analysis course should be a careful introduction to the real number line and functions of a real variable, based on rigorous proofs.

Typically, engineering students are required to take the sequence of calculus courses, but not required to take an analysis course. We generally push the stronger students to continue with the analysis sequence, and their "reward" for doing so is the opportunity to add a second major in mathematics, which has a non-trivial amount of employment value.

This system suggests that analysis is not required in a bare minimum of what a prospective engineer needs to know, but that it is a valuable asset that some engineering firms will value (either directly for the subject knowledge obtained or indirectly as a signaling mechanism to indicate an ability for abstract thought).

Answer 2

For one thing, you can argue that students need to know separation of variables http://en.m.wikipedia.org/wiki/Separation_of_variables and therefore need to know Series.

Answer 3

Technically the answer is "What an engineer (or any other person in a STEM-related profession) will need from an analysis course may be anything from the very basic understanding of the linear approximation and rudiments of the differential/integral calculus and rules for finding extrema to the rigorous grasp of the subject at a professional mathematician's level".

I would just make a clear outline of the course the way you want it, discuss it with the people from the engineering department, modify it according to their needs but preserving the internal logic and feasibility as you see them and, hopefully, get their support, so it will be they, who will argue with your department head, not you. Your only real argument can be that what you propose makes sense and hence is worth trying though, of course, your proposed course of action is not the only one that makes sense (and the same is the most that can be said of the suggestions of all other people, including your department head). That is the best practical advice I can think of.

As to the students, you owe them a clear explanation of what the course is going to be about, what the expectations and requirements are and other stuff like that but you are absolutely not obliged to take their wishes or shortcomings into account beyond the usual courtesy of spending some time in the beginning to refresh the required prerequisites in their minds. No matter what you do, you won't be able to hit just the right spot for everybody who might be interested in or forced into taking analysis class this year, so if your choice of how to run it is not completely outlandish and is supported by at least some good people in the engineering department, be sure that it is just as good as any other one with the same properties.

Ideally (in my humble opinion of a total ignoramus in the modern advanced pedagogical science), there should be several different analysis courses taught by different people with different attitudes toward both the subject and the teaching philosophy in general and the students should be allowed to freely select which one they are going to take themselves with no conclusions or, God forbid, penalties, drawn from the natural result that some classes will attract 100 people and some only 10 this way. We want to teach all 110 and we never really know whether that group of 100 or that group of 10 would be worse to lose because any real craft (and engineering is such) is moved forward by "mass effort" just as much as by "insights of the few", both being indispensable. However, if there is only one section with a single instructor offered each year, the best approximation to that would be to give him/her free hand (within the limits stated above) and rotate from year to year among the people who are interested in teaching the course (Yeah, I'm advocating for diversity here, though I'm afraid that my understanding of this word is somewhat different from its current mainstream meaning).

Just my two cents :-)

Answer 4

Engineers use Pascal's triangle and Taylor series approximations.

Can you relate sequences and series to error analysis?

The Fundamental Theorem of Calculus is necessary, so that they can check their work. Most Differential Equations courses (over-)rely on it.

Engineers need to be able to take derivatives and integrals of polynomials, exponentials, sine waves, and cosine waves in their sleep. And set up the boundary conditions for these problems, too.

Engineers should know how the limit formula for a derivative works, why every single term for it is what it is, and how to reverse the derivation to get an integral.

Consider other textbook authors, such as Swokowski.

Answer 5

1. It would be very unusual to use Apostol for engineers in the US. That book is used at Cal Tech (where A taught) but is very rare in the US landscape versus texts like Stewart, Larson, Thomas, Swokowski, etc. It is not used at decent engineering schools (but a cut below Cal Tech) like Navy or Georgia Tech.

2. Even at Cal Tech, that book and the course related to it are somewhat of a rite of passage and somewhat resented. For instance later courses, like ODEs, allow students the option to omit a heavy analysis theoretic emphasis and return to a more conventional approach.

See here, the choices for "practical" versus "analytical" flavored classes.

https://pma.caltech.edu/research-and-academics/mathematics/math-course-schedule-and-websites-2022-2023

[Copied for link rot]

Math 2/102 - Differential Equations

Fall Math 2 - Practical Dr. Makarov MWF 10:00 - 10:55 am KRK 119 Course Webpage

Fall Math 2 - Analytical Dr. Hutchcroft MWF 10:00 - 10:55 am Linde 310 Course Webpage

(For practical, they've often used Boyce and Diprima, i.e. a conventional text.)

http://www.its.caltech.edu/~matilde/Ma2aFall2010.html

[copied for link rot]

"Textbook William E. Boyce, Richard C. DiPrima, "Elementary differential equations and boundary value problems", 9th edition""

3. FWIW, I have a general engineering undergrad minor, have worked at an engineering firm (mechanical/electrical), passed the EIT, and almost took the mechE PE, and have worked in several industries with heavy engineering use, and even nuclear power and NASA. I think you have a very academic view of engineers (were at an institute and know engineers who taught). Don't assume you know what engineers need based on your very biased (statistical meaning) sample of them.

4. In general, working engineers barely use algebra. IN GENERAL. (But generalities are how practical decisions of allocation of effort are made...this is not a "divided by zero" real analysis exception situation, so please spare me some example of a Ph.D. engineer using real analysis for something.) They do need calculus and diffyQs to get through their majors courses (fluids, heat transfer, etc.) But in the work world, it's rare to even use simple pieces of calculus. There's a lot of adding heat loads or picking equipment from a catalog or sizing a pipe using a nomograph.

5. It's interesting that you feel outgunned because of your colleagues knowing more pedagogy. Well...maybe this is more of a pedagogy situation than a "how good at math are you" situation. In other words, the educator part, not the math part. Your colleagues are probably making arguments similar to mine (but based on experience with the actual students at your school). Instead of looking for sympathy (or arguments to support your opinion) on the Internet (and forums like this, or MSE, or Physics Forum, tend to attract outstanding people in terms of math ability who prefer harder books...after all "I can handle it"), you might want to study your students, analyze your colleagues remarks, etc. In other words, try to reflect and consider that you might be wrong.

6. For what it's worth, even the easier books in the US, usually have some theory. Topics like FTOC, Rolle's Theorem, Taylor series continuity are certainly taught. Just open the index of say Thomas...you will find pages with stuff on those topics. However, perhaps not with the intense rigor you want. All that said, even this theory is really just the bleeding in of analysis that has happened since the mid 19th century when it was developed and then bled into conventional texts over time (most heavily after WW2). I remember learning epsilon-delta in Thomas and enjoying the long pages of notation to work a few small problems. And then my teacher said, "that's great, but you're not going to need it the rest of the class". I was disappointed...but it turned out she was right. You don't epsilon delta to do some partial fractions or trig substitution or related rates head scratcher. And consider that Euler did some pretty badass calculus...and he died 30 years before Weirstrass "sprungt aus der Ei".