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My question primarily concerns the necessary transition from an undergraduate program in electrical engineering to graduate program in applied mathematics or pure mathematics.

I'm an electrical engineering student. During the first year in my university life, I found myself really fascinated with mathematics, and this summer after my first year of school, I self-studied Velleman's "How to Prove it", and analysis from Spivak's book.
As someone who had never been engaged in the circle of serious mathematics, I am lost as to the purpose of my studying: is it too late/highly improbable for me now to actually pursue a future in applied mathematics or pure mathematics while remaining in engineering as an undergraduate? Although I do have good reasoning skills, and finished Spivak's book in two months, I'm know I have much too long a way to go. Hence my question: should I try to take some mathematics courses outside my program such that I could partially fill the gap of my knowledge and basic abilities of mathematics? If so, is there any general area of math courses I should take? And should I actually complete a math minor or major degree (in my school specialist is ranked higher than major)?

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Yes you should try to do at least a minor in math if you are so inclined. I've always felt that the best engineering students I interacted with were those who had a solid foundation in (and appreciation for) math. A program you might want to follow is the analysis courses, particularly real analysis and complex analysis. These two will give you a solid foundation for understanding Fourier and wavelet transform theory which play a huge role in engineering. –  Cameron Williams Aug 5 at 2:55
    
I see, the real and complex analysis courses are indeed the two most recommended series of courses many weigh heavily upon! –  Eddy Aug 5 at 2:57
    
Many of the most successful applied mathematicians are those with a strong background in pure math but choose to pursue applied problems. Coming from an engineering background, you'll have a good appreciation for the applied side of things. The pure math side of things will help you further along your way to be a math-oriented engineer or an applied mathematician. :) (Note: some of the best applied math papers I've read were from math-oriented engineers.) –  Cameron Williams Aug 5 at 2:57
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Beware; Mathematics is a dangerous siren. –  copper.hat Aug 5 at 2:58
    
Aha quite true, although people around me have very mixed comments on my excitement about something outside my curriculum. –  Eddy Aug 5 at 3:01

3 Answers 3

You're probably in your early 20s, so: relax, you're okay. There's plenty of time to learn math. You have not missed some sort of mandatory train for becoming a mathematician or even just doing math at a reasonably advanced level. College is a great time to start doing mathematics.

If you finished Spivak in just two months (depending on the detail you went into), then you're probably already a great deal more capable than most of the students actually doing the math major. So don't think that you have too long of a way to go; you're obviously able and interested, and between the two of those, you can probably complete some sort of math curriculum pretty quickly. I know that I was in that situation as well, at one point. I ended up knocking out the entire math major in a single year by taking all core sequences concurrently. Caution: your mileage may vary.

If you can (i.e. if it fits into your program in terms of scheduling), I'd definitely recommend doing a math major. If you're looking into going into engineering, you must know linear algebra, so take that. Multivariate calculus is also recommended. Beyond that, real and complex analysis are also staples of applied mathematics, along with courses in differential equations.

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Well I did finish all the questions in the book, but that simply adds more agitation in my mind, because frankly speaking, the book contains really difficult problems; and as an "amateur", I just don't know what I should expect in higher-level courses, and programs. But thank you for your encouragement, I do lack confidence in myself, though that might be called discretion, soemtimes. –  Eddy Aug 5 at 3:05
    
@Eddy Finishing all the questions in Spivak's book is a serious achievement. It is not an easy introductory text. I assure you that you are capable of doing mathematics at an advanced level. –  Newb Aug 5 at 3:07
    
I see, then I think I will speak to my counselor about a math major in my university. Thank you for your great advice, thank you. –  Eddy Aug 5 at 3:11
    
@Eddy sure! good luck. –  Newb Aug 5 at 3:12

As an engineer interested in mathematics, you might want to look into the field of Continuum Thermomechanics. There are (applied) mathematics departments which offer such courses; yours might be such a school. Since you mentioned that you have done some self-study, books to look at as an introduction include:

1) The Mechanics and Thermodynamics of Continuua, Gurtin, Fried, & Anand

2) The Mechanics and Thermodynamics of Continuous Media, Silhavy

3) Many other freely available texts/sets of notes which are easily found online.

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Thank you for the recommendation! I will certainly go and see what they present. –  Eddy Aug 5 at 3:28

Aren't engineering students supposed to know algebra and calculus? While mathematics is certainly a better major than, say, the study of 15th century Armenian literature (which is fairly useless as far as today's jobs are concerned), you would be better off studying something with the greatest likelihood of landing you a good job after you graduate. In that regard, electrical engineering is better than pure mathematics.

If I were you, I'd concentrate on the math courses required for the electrical engineering degree, while cultivating the study of things like algebraic number theory as a hobby. Once you have a good degree under your belt and a steady paycheck coming your way, then you can start to explore more philosophical fields of study.

That's the advice I'd give my son.

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Indeed, what you are stating is one of the major reasons of my reservation as to what I should actually do with this late-coming passion of mine. However, I just do not want to have anything to regret for my years as an undergraduate, and since I am one of the top students in my engineering program, I think I will proceed and discuss the matter further with my counselor. –  Eddy Aug 5 at 3:24
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Yeah, definitely a counselor in person should give you better advice than some anonymous jerk on the Internet. However, one better: an in-person talk with someone who started out with an undergrad in math and went on to get degrees in engineering and open a major consulting firm. Maybe you can't talk to Ambassador Yousif Ghafari, but there's got to be someone like that who donates to your school. –  Robert Soupe Aug 5 at 3:29
    
Haha sir you do me too much honor, I learn from anyone older than me, as my major interest back in high school was philosophy (well that might not be as great a justification as I wished). I will certainly widen my eyes with as much effort as possible! –  Eddy Aug 5 at 3:33
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Bob gives good advice here. I would only add that you should read Paul Nahin's bio: amazon.com/Paul-J.-Nahin/e/B001HCS1XI I know of Nahin mainly because he's the author of at least two books on mathematical topics, but it turns out all his degrees are in electrical engineering. –  Alonso del Arte Aug 5 at 13:15

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