"For every equation you introduce, you cut your audience in half."
This expression, which I believe came from Stephen Hawking, summarizes why I believe that I have a weak foundation in mathematics.
I've always been fascinated by equations; their applications, implications, and histories. I would always wonder where equations came from and how they were developed, meanwhile losing track of the information that teachers were throwing at me. Unfortunately, this is still the case even now 😛
As an electrical engineering major/math minor, I feel obligated to understand the concepts that I learn in math at a fundamental level. Unfortunately, I didn't really respect math back in middle school and high school, so my understanding of certain topics in math are fuzzy at best. For example, I've passed Calculus I/II/III and Differential Equations, but I never really understood what I was doing; the calculations I'd make were by rote and not by intuition.
So I'll ask, where can I start? What do you all recommend I do? Any specific books/sources?
Thanks
Best Answer
Have you considered taking a class in "Advanced Calculus" or an (undergraduate) introductory class in mathematical anaysis?
It is often the case that Calculus I, II, III and Differential Equations are taught as "rules to apply" and "procedures to follow", so it is not unusual for students to feel as you do.
The "theoretical" understanding of Calculus is usually the goal of courses in Advanced Calculus and/or undergraduate mathematical analysis. If such a class is offered at your university or college, you might want to consider enrolling.
If you want to cover this territory on your own, I'd suggest previewing Serge Lang's Undergraduate Analysis, to see the topics covered, and if it looks "doable", obtaining it (perhaps you can find it at your campus library). There is an accompanying text: Problems and Solutions to Undergraduate Analysis, which I'd recommend, for self-study.
At the same time, try to obtain or borrow a copy of Walter Rudin's PMA: Principles of Mathematical Analysis (which has excellent exercises). You can obtain Prof. Silvia's COMPANION NOTES: A Working Excursion to Accompany Baby Rudin, to help work through the text on your own.
Here's another "self-study" option: See MIT's OCW Introduction to Analysis I. The text for the class is Rudin's PMA. Through the MIT site, there are problem sets to work through, lecture notes to accompany the text, and exams/quizzes available to test your knowledge. The prerequisites listed for the class are multivariate calculus and differential equations.
Note: If you find the above to be overwhelming, you might want to revisit Calculus with an emphasis on theory and concepts. For example, MIT has a course: Calculus with Theory, which uses the text by Tom Apostol: Calculus I, and a follow-up class: Multivariate-Calculus with Theory, which uses Apostol's Calculus I and II. The classes, together, cover calculus, differential equations, and introductory linear algebra, so you could follow the syllabi, lecture notes, work the problem sets, and in so doing, revisit most of the material you covered, but at a deeper level.