Introducing Rigor to my existent knowledge (Calculus and Linear Algebra)

[nunoxic]

I am an Mechanical Engineering major and just recently I found myself to be very interested in Computational Mathematics. I have taken the typical 3 courses on Calculus but the problem is that since they were being taught majorly to an engineering audience, no one bothered with the epsilon deltas. I know how to find limits, differentiate weird functions and integrate improper integrals but when I try to read research on Nonlinear Optimization or Numerical Linear Algebra, I am unable to follow the math because of its rigor.

I have some time to work on this before joining graduate school and wanted a recommendation for a good book that would allow me to revise the 3 semesters of calculus while simultaneously improve my ability to read and write rigorous proofs. However, I don't want a book like Apostol or Spivak (which run for thousands of pages) because I already know a lot of Calculus and honestly, it gets boring.

My goal is to finish revision and introduction of rigor as quickly as possible and move on to more advanced stuff that I am currently stuck on (Boundary Value Problems in Fluid Mechanics, NLA and Optimization to name a few)

I already have Apostol and Kenneth Ross but the former is tooooooo long and the latter (I believe) stops abruptly.

Any suggestions?

 

[Obis]

I'd suggest to search the internet for calculus lecture notes - they are usually more concise than books, covering the most important topics at the same time.

Another option would be to get a book similar to "Mathematical methods for scientists and engineers" by Donald Allan McQuarrie. Such books are one thousand pages long, however, they cover a lot of different topics, that are the most important to engineers.

 

[DivisionByZro]

I find your post a little confusing, you say that you have trouble following the more rigorous texts and yet you wish to avoid Spivak (Which does not run for anywhere over 400 pages, I don't have it around me right now). Spivak is very good at getting one used to more mathematically abstract thinking. You wish to learn epsilon-deltas, well Spivak is an ideal place to begin, did you even take a look at Spivak at your library?

In any case, if you want a rigorous linear algebra book, you could try Friedberg, Insel and Spence; or if you've taken some abstract algebra (most engineering majors around here don't) then Hoffman and Kunze is also great; lastly you could look through Hefferon's free linear algebra book, it is also quite nice.

Good luck!