Posting for my son (who does not have an account here):
He's a sophomore math major in college and is looking for a good book on Lie algebra and Lie Groups that he can study over the summer. He wants mathematical rigor, but he is thinking of grad school in theoretical physics, so he also wants...
I used the Piskunov volumes in my undergrad math curriculum at the Indian Institute of Technology (IIT) in the early 80s. Good books with lots of problems, but a bit dry. I still have my copies on my bookshelf.
He did say that he was just starting off, which is why I think that Spivak or Apostol would be too much. I really like the Silvanus Thompson book. I picked up a copy myself yesterday and it is really quite brilliantly written. It is an excellent introduction to the concepts.
Are you in the United States? If so, have you considered opening a Mu Alpha Theta chapter? I think my son did it in his school a few years ago (he is a senior now and about to go to college).
You can get details here:
I have not had personal experience with reading it, but a lot of people whose opinion I respect have suggested "Calculus Made Easy" by Silvanus Thompson. I flipped through it at the bookstore and it does seem like a nice introduction to the subject, emphasizing the concepts rather than the...
Apostol's calculus books are fantastic for a first course on analysis, i.e. for a SECOND phase on calculus in the standard pedagogical sequence for most people who want to study mathematics formally. They are too dense to be useful for a first course on single or multi-variable calculus.
If you are not from UK/canada/Aus/NZ, you will almost surely be required to take the TOEFL. Indians have to (at least we had to when I graduated in the mid 80s), and most Indians here have excellent English.