As an undergraduate, I took five semesters of calculus. The last semester on partial differential equations supposedly gave me a head start for grad school in my chosen field of chemistry.
But, unlike my friends in engineering programs, I had little need for calculus except in a few physical-chemistry labs and classes. I can't remember using calculus in grad school, and since then I can't remember using it more than a few times to solve electronics problems. Instead of having to take all those calculus classes, two semesters would have provided sufficient knowledge. Classes in scientific data analysis, instrumentation techniques, and statistics would have made more sense, but at the time either the courses didn't exist, or no one suggested them.
I looked at the math requirements for an engineering school, and found the following requirements for a BS degree:
Calculus I and II
Ordinary Differential Equations
Discrete Mathematics
Statistics
That seems like a better series of course requirements (one semester each) for engineers than the "exposure" I got in college. My biggest impediment to learning calculus centered on the lack of a relation between it and practical problems. After the math professors got past a few simplistic examples, it was all x-this, y-that, and everything between plus and minus infinity. Practical examples of why nascent scientists and engineers might need a triple integral might have helped us understand calculus better. Or perhaps I'm just more of a practical than theoretical mind.
We now have many computer programs that can help analyze data, and that provide sophisticated mathematical tools. And using such tools takes the drudgery out of tedious mathematical analyses and equation evaluations. So, does it still make sense to load engineering and science students with more than an introduction to calculus? If so, what level of calculus should they reach? (Of course, math majors and students interested in calculus could still take more courses.)
During college, I had a dull-as-dirt calculus text I was glad to throw out. Since then, I discovered The Calculus Tutoring Book by Carol and Robert Ash. The IEEE Press published this book in the mid 1980s, and although now out of print, AbeBooks.com lists many copies at reasonable prices. I recommend this book highly to students who struggle with calculus. One summer I went through about half the book on my own, and worked many of the problems. Finally, I found calculus interesting and clear. It only took an extra 20 years to "get it."
What level of calculus should engineering students reach? Discuss in the comments section below.
This is slightly off-topic, but I was recently reminded of something which happened when I was in college, in the first week of my differential equations class. One of my classmates had apparently managed to get through two semesters of calculus without learning any actual calculus. I'm not quite sure how he was able to do this, but he had a TI-89, and used it extremely proficiently. However, he had absolutely no grasp of any of the underlying math concepts. He literally only knew how to punch things into his calculator.
In the first week of class, the professor announced that there would be no calculators allowed on any of the tests. My classmate panicked. The look of horror on his face was visceral. At the end of the class, after the professor walked out, my classmate sat slumped at his desk in a state of shock. He never showed up to class again.
I never saw him in any of my other classes, so I strongly doubt that he stayed in engineering. Reflecting on the fact that he basically wasted a year of his life and presumably had to completely change his career goals made me feel better about having nothing but a cheap solar calculator. I also wondered whether things would have turned out differently for him if he would have put the same effort into learning calculus which he obviously put into learning how to use the TI-89.
I hope I don't sound like I'm ridiculing this guy, because I actually feel sorry for him. I strongly suspect that his overreliance on his calculator started in high school. One of his high school teachers should have identified this problem and helped him with it. By not doing so, they set him up for failure.
(Of course, I'm sure that some people reading this will argue that knowing how to solve calculus problems on a TI-89 is all the calculus you really need to know anyway).
Any subject, to be absorbed by a student, requires a good text and a good teacher. I have had my share of both in every possible combination of good and bad. My basic calculus text was obviously written to impress the author's peers, not to edify the students. My professor was good, but my progress wasn't great until the public library had a 5¢ a pound book sale and I came across Elements of the Differentialand Integral Calculus, by W. A. Granville (1904)—an ancient text, but presented so clearly, logically, and without assumptions, that you HAD to absorb and understand the subject. (This book has been of material assistance to two children, two grandchildren, and a high-school-teacher friend.)
I had a course in advanced engineenring math in which the instructor was so bad that we students ignored him and just read the fine text.
How much calculus? I believe that this should depend on you course of study. The minimum should be basic differential and integral calculus and differential equations. After that, it would depend on whether you're studying Chemical, Civil, Mechanical or other Engineering, sciences, etc.
Languages? The more you can. I think that these days, Mandarin Chinese is an essential. Have you noticed that everything seems to be made in China? Learning a language has the side benefit of exposing one to that language's culture and mental attitudes—a necessity if you're going to do business with those people.
All of this is aside from expanding your knowledge and mental flexibility. Isaac Asimov once said that the breadth of your knowledge is directly related to your creativity.
I graduated with a BSME in 1988. I had to take calculus I-IV and a separate differential equations course. Those were offered through the Dept. of Mathematics. Then I had to take a further 8 hours of "Advanced Engineering Mathematics" which was through the engineering department. It was all calculus. That's 23 hours worth. I've used it some in my career, but not nearly as much as they crammed into my brain.
While I don't completely disagree w/ your position, I for one am very grateful at having taken all those Mathematics sequences in college. As a "pre-engineering" major at a 4-year accredited college in the early 1960s, I was enrolled as a Mathematics major, w/ a concentrated minor in Physics. Not only did I take the entire Calculus sequence, but as other respondents have mentioned, I also took the advanced Algebra sequence, including ABSTRACT & LINEAR Algebra, and NUMERICAL ANALYSIS. In addition, I also have LINEAR & PARTIAL DIFFERENTIAL EQUATIONS & COMPLEX VARIABLES to my credit, and finally, a course of PROBABILITY & STATISTICS. All these courses, taken as a whole seemlessly meshed w/ the PHYSICS curriculum. There's no way that I could have been successful in all the PHYSICS courses, had it not been for the well-grounded basics of the Mathematics program.
As an accomplished electrical/electronic engineer today, I think I'd be less than successful had I not had a good foundation in these curricula. Furthermore, I'd probably NOT have been as effective as a "radio" engineer, had it not been for the course in COMPLEX VARIABLES, given the basic need to deal w/ capacitive & inductive reactance, admittance. It was always about "a+jb"!
About the only courses that have served no useful purpose to me was AN INTRODUCTION TO ALGEBRAIC TOPOLOGY. Ironically, I still have that textbook to this day.
This relates to a subject that has been bugging me: I think you need calculus just to read Wikipedia. It used to be a pretty good place to get a handle on new technology but now every page is filled with derivations and complex notations and it has become a place where you need a graduate degree in mathematics to read.
Taking four semesters of calculus was a lot of work for a skill I don't often need. Mostly, I just translate: integrate means I need an opamp and a capacitor, summation = an opapamp and a couple of resistors, differentiate = an opamp and....
I took a thirty-one week statistics class at work and the understanding and tools I got out of that are far more useful day-to-day in test & measurement.
At my school, Lwrence Institute of Technology, we had four levels of calculus plus one about differentials and matrices and a lot of that stuff, all taught by a brilliant professor who had it all in memory. The calculus was a great foundation for understanding physical reality, and it was also quite a hard class, as I recall, but not nearly as hard as my dynamics class was.
I once did watch a contract engineer calculate the exact heat sink required for a project using calculus on the waveforms of the pass transistor. I was even more impressed when the thing ran a constant ten degrees below the maximum allowable temperature limit. If we had been selling thousands of that instrument the 75 cents that he saved on the heatsink would have meant something, but for a one-off product it did not make sense.
In other areas the calculus was the foundation for setting up the equations, but the math was always just simple arithmatic, multiply and divide type of stuff.
A course in the rules associated with valid statistics would have been very handy, along with a handbook of methods. The class would need to be about the methods to assure that the results were correct. The incorrect application of statistical deviation data has caused a whole lot of problems, in my experience. The problem is that the results can not be more accurate than the noise level will allow the data to be. And data accuracy can't be better than the noise level. In other words, when you have a scope trace that is a quarter inch wide with high frequency noise, you can't expect to accurately read the center of the trace with a measurement curser.
You've got me there, Sparks. You make a strong case for greater breadth of knowledge. I have to admit, though, I never thought I'd bump into anyone who wished they had had more vector calculus.
"I do wonder about my vector calculus course, though..."
LOL, I'm an electronics engineer (an analog dinosaur) and I WISH I had a better vector calc background. Not for analog design, rather for orbital dynamics.
What do analog design, orbital dynamics and vector calc have to do with each other? Normally, there is not much at the crank-the-handle analog design level.But when called on to support an attitude control system on a spacecraft, there are elements of both worlds that co-mingle. The understanding of both makes anomaly resolution 22,300 miles from the anomaly that much easier. I speak from experience.
As for usefulness of foreign languages (or grammar skills or anything else not "engineering"), aside from the snootiness factor (IMHO, Americans are language deprived compared to their European kin), look at it like this: breadth of knowledge excites different areas of the brain. Much like practical music knowledge music enhances math and science skills, learning multiple languages expands the brain, the database engineers draw on to bring practicality to science. Problem solving uses different parts of the brain -- and the parts of the brain used appear to be dependent on what the problem is.My experience with truly brilliant engineers is that the smarter they are, the more their breadth of knowledge in areas other than engineering.
Thinking out of the box requires that there is some knowledge out of the box. If not, box barriers will rarely be breached.
I struggled with Spanish. Looking back it may have been my poor hearing, or then again perhaps the way my brain is wired. Language, in my not so humble opinion should not be a requirement. My favorite book over the years is "Calculus Made Easy", Silvanus Thompson, released in Dec 1911. Yes, over 100 years ago! A new and updated release is now in print, and in my old book I found several errors that I have corrected in the margin. It is very easy reading. I keep my old "Calculus Problem Solver" and use it mostly as a weight when I'm gluing parts together!
Still 90% of my math needs are met with algebra and trig. Stats I have used mainly in self promotion exercises, accounting for another 7% of my needs.The remaining 3% involve infrequent use of calculus at work or for blue sky thinking.What I really would like to do is understand Maxwell's work (not Heaviside's rework) and vision. (I'm not as smart as I like to think I am.)
I also believe in no wasted knowledge.While learning about logarithms I recall thinking, that this is a nonsense effort without any practical use.Today, in the RF field, I use it daily.Of course my TI-36X beats the hell out of the old paper log tables or my slide rule.
I agree with the idea that an engineer's education should contain a richer mix of mathematics. But calculus does train the mind to pursue other math disciplines, (on our own, if necessary) and I think that's one of the primary purposes it (should) serve in the undergraduate curriculum.
And don't forget that somebody had to master the numerical solution of systems of linear differential equations in order to make our CAD programs work.
One of the joys of graduate school is that you suddenly find yourself with the bakground and freedom to explore those other corners of math that you couldn't when you were deep into trying to understand thermodynamics and Maxwell's equations.
A story: I did my most interesting work with calculus after I quit doing engineering and started working on long-range planning and forecasting. You just never know when that stuff is going to be useful!
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