Expert: Abe Mantell - 8/23/2010

Question
Hi Abe,

I am an undergrad. physics student.
I have a bunch of calculus books and they are quite distinct from each other.

They are:

1.)Calculus by Tom M. Apostol
2.)Caluclus by Michael Spivak
3.)Calculus and analytic geometry by G. Thomas and R.L. Finney

The first two are almost the same in that there presentation is a "pure mathematical one" whereas the last book, seems to be an exhaustive course in applied calculus, stressing more on problems and less on formalities and rigour and proofs of theorems etc. and it's a book where one could find almost all application based problems from all the mathematical fields like engineering, physics etc...

I'd love to learn it from a pure mathematical point i.e. the first two books, but the drawback they would perhaps have is not give me enough skill or drill to deal with problems at a practical level, whereas the problem with the last book is, that it is not rigorous.

I also read at the caltech(California Institute of Technology - a distinguished school for physics) website, where under the heading of "what are you expected to know as undergraduates while applying for grad. prog." was mentioned a book on Analysis by the same author above , i.e. Tom M. Apostol.

Analysis, I guess is a pure mathematical subject but still they(caltech) expect that from us at the undergrad. level.

And so then, which of these approaches should I prefer? The pure math rigorous approach(Apostol and Spivak) or the applied approach(thomas and finney)?

Thanks

Answer
Hello Shikhin,

I am familiar (and have) all three of those texts, and it seems to
me that you have a good understanding of the pluses and minuses of
each approach.

Personally, I like both...I like to understand mathematics from its
roots and know why it works, but I also enjoy applied mathematics and
like to see how it can be applied in the "real world."

My advice is to do both! Especially for you, a physics major, I think
you'll greatly benefit.  If you were an engineering major, then I'd
say you probably do need the mathematical rigor of the Apostol or
Spivak text, and should focus on the practical use.

I think physics (usually) demands both mathematical rigor to some
degree and, of course, mathematical applications.

Do you have the time to study the two different approaches?
If so, then perhaps do it in a parallel fashion.  Read a chapter
from Spivak or Apostol, then after you get the mathematical rigor,
look at the correspondning chapter in Thomas/Finney.  I think you'll
find the applications are pretty easy by comparison (but interesting)!

Good luck!

Abe