I'm an electrical/computer engineering student and have taken fair number of engineering math courses. In addition to Calc 1/2/3 (differential, integral and multivariable respectfully), I've also taken a course on linear algebra, basic differential equations, basic complex analysis, probability and signal processing (which was essentially a course on different integral transforms).
I'm really interested in learning rigorous math, however the math courses I've taken so far have been very applied – they've been taught with a focus on solving problems instead of proving theorems. I would have to relearn most of what I've been taught, this time with a focus on proofs.
However, I'm afraid that if I spend a while relearning content I already know, I'll soon become bored and lose motivation. However, I don't think not revisiting topics I already know is a good idea, because it would be next to impossible to learn higher level math without knowing lower level math from a proof based point of view.
Best Answer
Consider going through Calculus by Michael Spivak or Introduction to Calculus and Analysis (Volume I) by Richard Courant and Fritz John. You may initially think you know most of the material in these books (because you can differentiate and integrate some standard functions, etc.), I think if you really hit these books hard by reading for understanding all the proofs and attempting as many of the exercises as you have time for, then you'll find they contain quite a bit that you are NOT familiar with. Given your engineering background, Courant would be my pick for you. See my answer to Difficulty level of Courant's book. See also the comments here.
Another suggestion is to get one of the comprehensive advanced calculus texts from about a generation ago, one that includes a rigorous review of elementary calculus before launching into an extensive coverage of sequences and series, vector calculus, elementary differential geometry, possibly some complex variables, etc., such as Advanced Calculus by Angus E. Taylor and W. Robert Mann, or Advanced Calculus by R. Creighton Buck. In past generations, the 2-semester sequence courses out of such a book tended to be the primary transition (and weed-out course) for undergraduate students to transition from elementary calculus and ODE's to upper level mathematics. Because the mathematics curriculum has gotten fuller in the past few decades (more discrete math, probability, and previously non-existent courses in the emerging discipline of computer science), these 2-semester sequence courses have gradually been phased out and replaced by more targeted 1-semester "transition to advanced mathematics" courses that have much less depth and far more focus on mathematical grammar issues and basic proof methods than the earlier advanced calculus courses, plus the "transition to advanced mathematics" courses are typically taken in the U.S. during one's 2nd undergraduate year rather than the 3rd undergraduate year in which the advanced calculus courses were typically taken.