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In my experience, many introductory engineering mathematics textbooks these days tend to skip proofs and discuss logic only in the context of digital electronics. On the other hand, I can imagine that engineers (and others) could benefit from developing basic skills in mathematical thinking* beyond the cookbook approach. (See, for instance, Keith Devlin's Introduction to Mathematical Thinking course.) I hasten to add that many engineering mathematics textbooks do have strong points, such as numerous examples of mathematics applied to solving engineering problems.

I would be very interested in learning about either engineering mathematics textbooks that do contain material on mathematical thinking or your experience(s) in teaching introductory mathematics to engineers where you went beyond the cookbook approach. Input from engineering students and practising engineers is also welcome, as are contributions from those involved in other fields in which mathematics is applied.

(For the record, I currently use Engineering Mathematics: A Foundation for Electronic, Electrical, Communications and System Engineers by Croft et al. The material covered includes derivatives, integrals, complex numbers, matrices, differential equations, Laplace transforms and Fourier series.)

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  • $\begingroup$ Subjects? For vector calc. I'd recommend Vector Calculus, Marsden, Tromba for one with examples and such, but another great one (my personal favorite), sliiightly less for engineers is Calculus in Vector Spaces, Corwin, Szczarba. $\endgroup$ – GPerez Jan 11 '14 at 14:00
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    $\begingroup$ While I am very interested in the answers to this question, I think this caveat is important: most can't see beyond the one-semester horizon. It is likely to be attacked with barrage of questions like "Will this be on the test?" or "What are real life applications of this? Will I ever use this in my life?" Sometimes I feel these questions are rhetorical, they don't expect an answer, they just expect to fool themselves that it isn't important. $\endgroup$ – Mark Fantini Jan 11 '14 at 17:42
  • $\begingroup$ @GPerez: Thanks for the suggestions. I've edited the question to include the textbook I use at the moment, along with some of the main subjects. (Please note that the list isn't exhaustive). $\endgroup$ – J W Jan 11 '14 at 17:47
  • $\begingroup$ @Fantini: Good points. Of course, it might be possible to include such material on the test, with suitable care. For applications to real life, perhaps this blog post by Keith Devlin might shed some light. $\endgroup$ – J W Jan 11 '14 at 18:09
  • $\begingroup$ You may want to look at Gilbert Strang's Introduction to Applied Mathematics. I have no idea how well it would fit with what you're teaching, but it could be a useful text to have on your shelf for reference. $\endgroup$ – Dave L. Renfro Jan 13 '14 at 16:33
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The assumption of this question, that engineers do not have the "basic skills in mathematical thinking" is wrong. Engineers have the same three steps (that Terence Tao has described in his blog) in their own fields. A simple example for the special case of an electronics engineer might be the following: Pre-rigorous stage: Have a simple light-bulb circuit, perhaps with a switch, imagine a current flowing, you open the switch the light-bulb goes off, etc, simple experiments even with transistors perhaps. Rigorous stage: Kirschoff's law, transistor input-output relationships, operational amplifiers, etc. Post-rigorous stage: I once witnessed the following. A huge circuit does not work for some reason that we could not figure out. A seasoned professor came, had a look for a few seconds, and just suggested, "throw a capacitor in there," and surprise surprise, it worked. He did not bother himself writing down the equations and solving them.

In this context, the common misconception of most ($\textbf{Edit:}$ "most" is not the right word here, let's say "some") mathematicians is that they think engineers need to know where Kirschoff's law is coming from a deep mathematical perspective (whatever that is), so that they will e.g. be able to analyze complicated resistor networks constructed via infinite superpositions of $K_5$ graphs. Believe me they don't. They are also usually not interested in the existence of non-measurable functions etc., because (usually) they do not need it. A similar analogy is that a number theorist, despite being a mathematician, perhaps does not bother herself with the existence of natural numbers (I am not a number theorist, so this may be wrong).

On the other hand, I also know engineers who know a lot of "high-level" mathematical tools, simply because they need those tools.

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    $\begingroup$ There are a lot of engineers out there. I know some who are highly trained and highly skilled mathematically. I have met some engineering students who are certainly not. I'm not sure how the field vets the level of mathematical knowledge of its practitioners, but surely there is enough of a range so that for any reasonable definition of "basic skills in mathematical thinking" some engineers have it and others less so. Whether they do or don't doesn't seem to be the crux of the question so much as that engineering mathematics textbooks don't teach it well (which seems to be true). $\endgroup$ – Pete L. Clark Jan 14 '14 at 2:54
  • $\begingroup$ By the way, what you say about number theorists is certainly correct. If I bother myself with the existence of natural numbers -- which beyond acknowledging that there's something out there called the "Peano axioms" I never have -- I am not (I feel) acting as a number theorist but rather dabbling in mathematical philosophy. $\endgroup$ – Pete L. Clark Jan 14 '14 at 2:56
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    $\begingroup$ @PeteL.Clark Thank you for your comments. Regarding textbooks, what I meant to imply is that for example, you do not need a measure-theoretical treatment of calculus in an engineering textbook. Or, regarding OP's specific question, you do not need to treat Boolean algebra in an "extremely formal manner" to teach somebody digital electronics. Most engineers (even the ones who are in academia) do not need to know about most of these foundational matters. $\endgroup$ – Lord Soth Jan 14 '14 at 3:01
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    $\begingroup$ In what calculus textbook does one meet measure theory? One typically doesn't even meet measure theory until graduate level analysis. Similarly I don't know what most mathematicians think about Kirchhoff's Laws, but the idea that some kind of foundational knowledge of graph theory would help you fix circuits sounds like thinking that learning game theory in the von Neumann sense would make you a chess master. I don't know many, if any, mathematicians who believe things like that. So some of your arguments seem "straw man-ny" to me. $\endgroup$ – Pete L. Clark Jan 14 '14 at 3:06
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    $\begingroup$ The quote said "mathematical thinking" and you said "analytical thinking". Those don't sound exactly the same. It is perhaps not sufficiently clear what the OP means by "mathematical thinking", but it seems like it is intended to mean a more conceptual understanding of mathematical concepts rather than an emphasis on techniques, calculations and memorization. Assuming that's what the OP means, are you insisting that engineers have those skills? All of them?? To the maximum useful amount??? $\endgroup$ – Pete L. Clark Jan 14 '14 at 3:18

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