Schools College calculus self-study question

[Alcubierre]

Hello, I am a recent high school graduate and I have recently finished calculus II with one year of calculus I prior (which I took junior year). My major will be physics at the University of Texas and having told that I completed those courses to my advisor he said I seemed bright from the conversations we had and wouldn't mind placing me in multivariable calculus. It is now summer and I have decided to study the course ahead. I looked for a PDF of Apostol's but I found it very rigorous in its presentation. I easily grasp concepts and methods and I can usually undertake difficult problems but some of the problems were extremely difficult in the sense that if you read the chapters and understood as far as the text goes, the problems seemed still too hard. So as for my questions: 1. should I study Spivak before Apostol? 2. What kind of mathematical background would one require to master Apostol? And 3. What is meant by "mastering" a textbook? To study as I plan to, is it a good idea to try to do every single problem or just conceptual/exploratory problems with some mechanical (to get used to the mechanics of problem solving) problems?

 

[MarneMath]

I'm having a bit of confusion. Your adviser has stated he would place you in multivariate calculus, so you want to study the course, which is reasonable. What I am confused about is why are you trying to go to Spivak or Apostol? If your intent is to learn multivariate calculus for a typical Calc III course, it would be more beneficial to learn from a more cookie cutter book, instead of trying to learn calculus I and II again from a more proof based perspective.

As for the problems in Spivak or Apostol, they are hard for a lot of beginners because proof writing is a skill that takes practice. However, most of the problems are do-able, with a few that are truly skull crushers. The jump from "oh this is obvious" to "why it is obvious" is surprisingly difficult for a lot of people.

 

[Alcubierre]

To clear up your confusion, my intent was to study calc I and II with proof hence those textbooks, because I feel that the AP curriculum (advanced placement) isn't so strong when I look at some exploratory exercises from those textbooks about topics that I am familiar with.

 

[Alcubierre]

I can't edit because I'm on my phone but to add on to that, what would be a recommended cookie cutter book?

 

[MarneMath]

Well, I tend not to recommend cookie cutter calculus books because they're essentially the same, so it would probably be best if you find out what book your school uses (I believe it is Stewart) and try to find a used cheap old edition copy and work through the Calc III material so you become familiar with it.

Going back to your first post. A strong mathematical student with desire to learn proofs is all that is really needed for Apostol or Spivak. Although it is advisable to have been introduced to Calculus already. Technically speaking, you should be able to handle Spivak or Apostol. I personally find no real difference between the two in term of difficulty. I enjoy Spivak's writing a lot more and find it easier to read than Apostol, but I think Apostol definitely has his own insights. So what it comes down to is preference. Odds are if you find Apostol hard, you'll find Spivak hard too.

 

[Alcubierre]

I didn't find Spivak difficult, just Apostol was too different in the way he presents the material. But thank you very much, I will definitely look into Stewart.