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"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 :P

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

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This is how most students are. You will understand it more when you do more advanced math. For example, as already mentioned in an answer, real analysis. Real analysis is where you prove calculus stuff. So, you will understand everything better if you take that. –  Graphth Dec 15 '12 at 23:13
    
Books there are aplenty, and you can look at the questions in the "Related" category to the bottom right for those. But as every professor of mine has told me, you don't learn math without doing math. Solve problems, read examples, imagine counterexamples...this is how mathematicians really learn their stuff. –  neuguy Dec 15 '12 at 23:15
    
I recommend Lawvere & Rosebrugh's "Sets for Mathematics" for starters, but really there's 1000 books one could read and still not know anything. –  alancalvitti Dec 16 '12 at 1:21

2 Answers 2

up vote 7 down vote accepted

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.


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I love Principles as much as the next person, but for someone who want to strengthen foundations I would suggest Calculus by Spivak before moving on to Principles. –  neuguy Dec 15 '12 at 23:19

It really depends on what area you want to understand. As you will realize if you continue studying it, mathematics is arguably the most specialized subject.

Anyway, if you want to understand the principles behind what you were doing in Calculus then you definitely need an introductory course in Real Analyisis.

Some books that I really liked and do not require almost any prior knowledge:

  • Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis
  • R. P. Burn, Numbers and Functions, Steps into Analysis (this one contains many problems and answers, the best way to learn mathematics is to do it!)

Then, when you have a decent knowledge of this basic material you could try to read one of the great classics of analysis:

W.Rudin, Principles of Mathematical Analysis

You might find this material really different from what you are used to, but this is the difference between mathematics and engineering!

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