A question from an engineering undergraduate 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)?

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. 
