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I did a minor in mathematics a couple years ago and the non-engineering (i.e. rigorous) math I have been exposed to were two proper courses in prob and statistics, 2 courses in real analysis and 2 courses in abstract algebra, the second one covering mostly linear algebra (dual spaces, span, Cayley-Hamilton, Jordan etc.) Now I'm in EE and we get thrown a bunch of stuff at our faces and I seem to be the only one being like, hey, guys, where are those formulas coming from. So I would like to carry on getting a good underlying understanding. For example the Laplace and Fourier transforms are used extensively, as well as the dirac and other generalized functions. I'm having courses in signal processing and control theory right now and have found while talking to profs that having a good math background would be a great asset to get research internships as the other guys have no idea what analysis is for the most part and would be lost in a more advanced context. So I am trying to turn that year in pure math into an engineering asset (also, hey, I like to understand stuff I manipulate everyday), be it not the most in depth knowledge of the different topics(e.g.category theory) I would like to get solid ground in the following:

  • Real analysis: I have read Bertolt and Sherbert cover to cover back in math but I think I need more now

  • Operator theory and distribution theory which I think would be included in any advanced real analysis book along with measure theory and maybe a tiny bit of topology

  • Chaos theory and non linear systems stuff for control systems. For example linearization of odes for state space representation

  • Furthering the algebra can't be bad, but is Dummit and Foote really the only way to go, I got to admit it intimidates me

By the way I realized I won't have time to grasp all that in the month, I am only trying to come up with a list of references as I get more wtf moments. (the most urgent being the distribution stuff, seriously, in EE we basically spend all our time playing with the dirac) Also I know the question for self studying is redundant and has many duplicates however I think my specific case has the following characteristics which make it relevant : transferred to EE, EE has some very specific mathematical tools that it needs (e.g. once again distribution theory, when people don't even grasp complex numbers is already introduced via circuit theory) also I would like my learning to be structured in such a way that it backs up the tools I use as I progress and use more subtle stuff. For example right now in control theory linearization of state space is simply done by taking first derivatives (giving Jacobian matrices), so understanding the distributions would be more urgent

Thanks a lot

Harish Chandra Rajpoot
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SolipsistElvis
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  • I think a lot of very successful EE phd students know almost zero about abstract algebra (except for linear algebra, which they know well). – littleO Oct 03 '15 at 04:08
  • It really depends on your goals. Engineers, in general, are focussed on using mathematical tools rather than dealing with proofs and justifying assumptions. This, in my opinion (which may be rather twisted), is mostly a matter of efficiency, you have only so much time. Your background will be invaluable in understanding, but may be a hindrance in that you will find yourself doing things that no self respecting mathematician would do. For example, few engineers know what a tempered distribution is, but will happily compute using the Dirac distribution. – copper.hat Oct 03 '15 at 04:08
  • @copper.hat Apparently you are not familiar with physics and engineering curricula. Both require a deep understanding of mathematical theory and the ability to develop proofs of hypotheses. So, please, do not relegate these disciplines to "use of tools." That is not true. Some of the best mathematicians are scientists. Ed Witten won the Fields Medal after all. – Mark Viola Oct 03 '15 at 04:13
  • (cont.) Also, different folks have different characteristics and interests. For example, I am almost incapable of dealing with abstract algebra (category theory, for example), but am comfortable with linear algebra/functional analysis. – copper.hat Oct 03 '15 at 04:13
  • Did you take courses in coding theory, control theory, communications, electromagnetic field theory? All of those require a level of mathematical maturity at which one can develop hypotheses and rigorously mathematically prove them. And there is a branch of physics called "mathematical physics," in which mathematical rigor is applied to conjectures in physics. – Mark Viola Oct 03 '15 at 04:18
  • @copper.hat Perhaps things have changed since I was a student and subsequently a professor 1,000 thousand years ago. – Mark Viola Oct 03 '15 at 04:19
  • @Dr.MV: Of course. However, for example, one uses tools like the divergence theorem, etc, without ever discussing integration on manifolds, Stokes theorem, etc. – copper.hat Oct 03 '15 at 04:19
  • I find this quite funny. I had the intuition that this question woud spark that whole engineering vs mathematics feud. For the story I remember that while in math some graduate students helping out newbies would refer to week proofs or lousy logic as an 'engineering math' But while I got a hold of an EE phd. @Dr.MV don't you think getting a better grasp of the dirac through real analysis would be a good idea? I have seen profs starting to look at me in another way since I've been namedropping some pure math and maybe help with recommandations for a phd in the future. – SolipsistElvis Oct 03 '15 at 04:20
  • Sorry, I don't want to stir up anything, I don't have the energy :-). I really think that, broadly, the focus is different, and there is a difference between using and developing tools. There are lots of counterexamples to such a simplistic statement. – copper.hat Oct 03 '15 at 04:22
  • @Dr.MV for example in signal processing I find myself quite dumbfounded by some results like reconstructing signals and derivative of the dirac and so forth. I feel like some level of rigor would ironically provide me with intuition through my problems – SolipsistElvis Oct 03 '15 at 04:22
  • @SolipsistElvis Physicists and engineers who are ignorant of Gelfand & Shilov are doing a disservice to themselves and their professions. Most of the ones I know are versed in functional analysis. – Mark Viola Oct 03 '15 at 04:23
  • @Dr.MV: I think you must work with a select lot... – copper.hat Oct 03 '15 at 04:23
  • @copper.hat well probably but hey, shoot for the stars right? – SolipsistElvis Oct 03 '15 at 04:25
  • @copper.hat I don't believe they are "select." They are competent and mature. By the way, as a former professor, the undergraduate curriculum was a bit light. Perhaps this is the distinction and a valid one. IMHO the average undergraduate physics/engineering student and the undergraduate math student do focus differently as you suggested - more of less. But there were a group of physics and engineering undergraduate students who would challenge hand waving interchanges of operations and other formal developments. – Mark Viola Oct 03 '15 at 04:27
  • @Dr.MV haha I'm quite often that person and people don't seem to appreciate that, were you reffering to that book: http://www.amazon.com/Generalized-Functions-I-Properties-Operations/dp/0122795016/ref=sr_1_4?s=books&ie=UTF8&qid=1443846455&sr=1-4&keywords=Gel%27fand and if so could you recommend any alternative, seems to be out of print and 345$ seems quite steep... Thank you btw to take the time to help out an undergrad, appreciate it – SolipsistElvis Oct 03 '15 at 04:31
  • You're welcome. My pleasure! There are many books on the topic of GF. Here are a couple of primers that I posted herein and here. – Mark Viola Oct 03 '15 at 04:35
  • @Dr.MV: This is a discussion better held over a glass of wine. – copper.hat Oct 03 '15 at 04:37
  • @copper.hat indeed, and a proper chalkboard, gotta say haven't found many people to share such a moment with yet though – SolipsistElvis Oct 03 '15 at 04:40
  • @copper.hat Joe, ... wine or perhaps even better Long Island Ice teas ... ;-)) although that might not be so good for my Multiple Myeloma. By the way, our experiences differ a bit wrt rigor in physics/engineering. Mark – Mark Viola Oct 03 '15 at 04:40
  • @Dr.MV: Sorry to hear about it. As an aside, I'm transferring to the wine phase of the evening :-). – copper.hat Oct 03 '15 at 04:46
  • @copper.hat Joe. Thank you. I am doing well for now. – Mark Viola Oct 03 '15 at 04:47
  • If you have lots of extra time, I would not use it that way. You can ask faculty if you can get involved in some research project. That will ultimately help you more. You can learn as you go as needed. One mistake students make is thinking they need to learn all the theory before actually working on something. – Michael Oct 03 '15 at 05:06
  • @Michael I agree with you. I am already in an electric car project which is making me learn tons on "real world" engineering with embedded systems and HV systems and the challenges of integration, pcb layout etc.(http://eracing.polymtl.ca/ we are looking for sponsors btw if anything...) and am sitting on my college's library coop's board. Math just keeps pulling back I guess. It's also the way my mind works, I have a hard time diving in without understanding which I undestand can be a liability. However I also try to see it as an asset and take time to understand on top of therest.but its hard – SolipsistElvis Oct 03 '15 at 05:21
  • Like, I have had this Ogata's control book open for a long time without running through it, just reading about analysis, guess I should go back to that – SolipsistElvis Oct 03 '15 at 05:22
  • You can also do things like take ballroom dancing or art classes or cooking and so on. But sounds overall like you are doing well. – Michael Oct 03 '15 at 05:32
  • haha don't worry, I have a vocal jazz class and am not introverted and sad if that is what I made myself sound like. In fact I have recently began treatment for OCD which was never diagnosed until recently and now feel like I have the energy for a million tasks (went from a slacker stoner to Leslie Knope). Also had a gap year biking Europe and smoking weed in communes and doing e with new travellers. So now I'm geared towards accomplishing stuff – SolipsistElvis Oct 03 '15 at 05:36
  • Also swing lessons with the girlfriend of the time. The relationship ended after 4 lessons which is unfortunate because I really liked swing lol – SolipsistElvis Oct 03 '15 at 05:37
  • wait sorry, forgot this isnt reddit – SolipsistElvis Oct 03 '15 at 05:43

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If you want a background in algebra check out "Algebra 3rd ed" by Saunders Maclane.

AnotherPerson
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