Jon, I completely agree. I took only 2 semesters in college, but that was plenty.Where you were able to recall at least a few examples where you applied Calculus during your career, I look back and cannot recall even one instance.While the subject matter was valuable to broaden my overall foundation for engineering, the practical application for me was lost.My vision of applied calculus was for NASA engineers who designed missions to Mars; a far cry from my day-to-day in electronics products.
I remember watching Tom Hanks playing Jim Lovell in Apollo-13 and recall the scene where he was busily scratching out equations that may well have saved his life – I thank God that my life never depended on my calculus skills, or I wouldn't be writing this today!
When I was working as an engineer, one of the engineers in our group needed to calculate the area under a curve. He asked for help, and pretty soon there were six of us around his desk, arguing about how best to calculate the area. The point is, calculating the area under a curve is one of the most basic calculus tasks, and most of the engineers in my group (who were all out of college for a few years) couldn't remember exactly how to do it -- at least for this particular problem. I took three semesters of calculus, one semester of differential equations and one semester of vector calculus, and never used it, at least not directly. I know that it's important for engineers to understand how equations are derived, and calculus is often part of that understanding, but it really does't have much bearing on how good an engineer is.
My high school sophmore son just asked me this question. His older brother, who is a freshman in Aerospace Engineering, took lots. That was in high school. He took AP Calculus B/C and then, as a senior, took multivariate calculus. They call this Calc 3 in our schools. Actually, it is taught at the University of Illinois at Urbana-Champaign. So, he has a college transcript before getting into college. He could have started with differential equations, but decided to take the equivalent of Calc 3 at his university.
Is this all really necessary? I have used it many times in my career. I am not using it now, but I probably use it in some projects in the near future. It is good to have a basis in it. It gives you more flexibility in future opportunities.
Your proposed basic program is very good. I personally would have taken lots more math, but that is just me.
I don't solve differential equations on a regular basis, but that's not necessarily the point. Hopefully, after several years of struggling through calculus and differential equations, a student will come to have an intuitive understanding and appreciation of the mathematics which governs the world around us. It's that understanding, much more than the ability to solve the equations themselves, which is necessary in an engineering career. Studying math also helps develop the discipline and reasoning skills which are needed to solve engineering problems.
When I was in community college, I took three semesters of calculus, one semester of differential equations, and one semester of matrix algebra. That was the extent of the math courses which were offered, but my calculus professor, Dr. John Wegner, gave an informal class to a small group of students in which he covered some more advanced topics. As far as I know, he taught this class for free. I am grateful to him for his time and dedication.
As soon as I transferred to a four-year university, I was eager to take a more advanced math class. I chose a course in complex variables. This turned out to be a mistake; as an engineering major straight out of community college, I was not quite ready for a course usually taken by fourth-year math majors. I managed to struggle through the class with a C, but really I understood very little. It was, however, a good lesson in my own limitations. (I still have the book, and am hoping to give it another look some day).
Later, for some advanced physics classes which I took as electives, I had to learn a few things about tensor analysis and calculus of variations. I had much more luck with these subjects than I had with complex variables. Calculus of variations, in particular, is a fascinating topic, which allows you to do all kinds of things you can't do with ordinary calculus.
Without a doubt, the most important math subject which I didn't learn in college is statistics. I took a "probability and statistics for scientists and engineers" course, but the professor resented having to teach non-math majors, and as a consequence, did a horrible job. Of course, I can't put all of the blame on the professor, since it's a student's responsibilty to learn, regardless of the quality of the teacher. But I essentially had to teach myself statistics several years later, when I was unexpectedly thrust into a job as a quality manager.
Since graduating, I've also studied a number of mathematical topics such as logic, number theory, and graph theory -- not necessarily because they're particularly useful to me as an engineer, but because they're interesting, and keep my brain cells from atrophying.
How many math courses should be required in an engineering curriculum? Given all of the other material which needs to be packed into four years, it's hard to say. Certainly one outcome of an engineering education should be a solid understanding of the mathematical principles which underlie the world we live in, and calculus and differential equations are essential to that.
Might one consider all the calculus courses as indirect training for problem solving? Granted you might not use calculus directly in most work, but the learning process, and all the homework did train you to solve problems, right?
I went through the same sets of classes, but I'm not ready yet to dump them.
I remember a physics lab in which my partner and I created a strip-chart recording that plotted a variable of some sort. Then we had to find the area under the curve. Because this was experimental data and not a function we could describe with an equation, we carefully cut out the "area" under the curve and weighed it on an analytical-type scale. Then we weighed a 10-by-10-cm piece of the same chart paper. After that, the "integration" was simple math.
Ebay has planimeters for sale from time to time, so one of those would do an integration of experimental data, too, but I don't know the accuracy of the results. Wikipedia has a short discussion about planimeters. Of course it uses integrals and partial-differential equations, so the underlying calculus is never far away.
Whether or not one actually uses calculus heavily throughout one's career is I think irrevelant because there are two factors I'd give greater importance to. One, mathematics is one of the foundations of engineering, so it's incumbent upon people who want to be engineers to learn math. (Eye surgeons still go thru anatomy classes and learn about the heart, lungs, leg bones, etc.)
Secondly -- and this is where I have a real bug in my bonnet (to mix metaphors so I don't write anything too sour) -- if you want to call yourself an engineer, you should have to make it through engineering school, and that means taking classes which are hard to pass. The PC and the Internet has killed the little respect the profession had by enabling every idiot who's written an Excel macro (or even a line of HTML) to think that they're technical people who are "engineers."
I know we can't put the genie back in the bottle, but I think we should try to keep up some barriers to entry, at least in the "real" engineering world.
I know engineering and science majors must take difficult courses, and we all have. But I draw the line at courses that have no practical value in terms of knowledge or techniques. Years ago science and some engineering students had to take a foreign language for 2 or 3 semesters. Supposedly those courses would help the students translate scientific papers from German into English. I took two semesters of German as a sophomore and one more semester as a senior. I used my "knowledge" of German exactly once. At the time a MS in chemistry required passing a translation test, but examinees could use a technical German-English dictionary! I never used German again and can't say I feel constrained without fluency in a spoken language other than English. As far as I know, most colleges and universities have given up foreign language requirements for science and engineering students, although a few might hold on. Students would do better to learn C or LabVIEW.
I agree that, with the sheer volume of material which needs to be covered in four years, it's unrealistic to require science and engineering students to learn a foreign language. But it ought to be strongly encouraged. Being bilingual in English and Spanish has greatly enriched my life, both professionally and personally. (For one thing, I doubt that I would have married my wife, who is from El Salvador, if I didn't speak Spanish). I wish I knew French and German, since I regularly work with colleagues in Quebec and Austria. Of course, most educated people in those places speak English, but if they went to the trouble to learn my language, why shouldn't I do the same? Mandarin Chinese would be a good one to know, too. I've met a number of second-generation Americans who seem to be proud of not knowing their parents' language. This seems to me to be a foolish attitude. The more languages a person can speak, the better.
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