Expert: Alon Mandes - 8/23/2010

Question
Hi Alon,

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 myself would rather prefer the pure mathematical approach, for
which you will master the art of calculus . It will then make it
easier for you to apply the gained knowledge to practical issues ,
such as physics and engineering .
The pure approach will provide you with tools and a very constructive
point of view regarding analytical approach .
In my own opinion , its very important to master most of the proofs
in calculus study. This will shape your way of analyzing problems .
It's always better to learn the pure origin and then the applications.
This is however, my point of view . I my self a mathematician %26 also an engineer .

Best of luck,
Alon.