It's been years since I took calculus, and while I have a good understanding of the theorems of single variable calculus from my real analysis courses, computationally I am a bit slow. It takes me longer than it should to compute basic derivatives and integrals, and I don't really remember how to do all the standard problems from differential/integral calculus (related rates, volume of a solid of revolution, etc.). I can typically derive the needed solution if necessary but I'd like to start doing some private tutoring and so I'd need to be able to explain the problems at a level the students would understand.
Also, I have some gaps in my knowledge of multivariable calculus, as I took it seven years ago and have not used much of it since then. In particular I don't remember anything about line integrals, surface integrals, Green's theorem, flux, divergence, etc.
What are some good books/references I can use to brush up on this material? As far as my background, I have a master's degree in a math-related field and took PhD level courses in real analysis, operator theory, probability theory, and stochastic processes, so I would like to be proving the theorems as I go along. But I also need to do plenty of example problems like you would see in an undergraduate calculus sequence so I can be quick at solving them.
Best Answer
What you are telling is that you probably forget almost all the advanced undergraduate materials in both algebra and analysis. You have a big gap to fill at this point. You could restart your learning by reading an intro to analysis book along with algebra or linear algebra. I guess that your algebra and linear algebra core knowledge content is long gone. I also propose that you make time to attend a lecture in either of these courses at a university so you can pose questions and get immediate live answer back. Don't try to learn on your own.
There is an old saying that you could liken it to an adage: " a teacher should not give too much wisdom to his students ". I agreed. But in this case, I am willing to go the extra mile.....
Again, base on your inputs, you need to find an advisor who can tell you what to do to get your terminal degree in the shortest possible time frame...