Why Is Spivak's Calculus Textbook So Highly Regarded?

[Ian_Brooks]

I honestly enjoyed using James Stewart (yes heathen) when I did calc I, II, III and now these forums are buzzing with Spivak - as if he's the new Giancoli/Feynman of math textbooks

whats the big deal?

Oh - and i did check out the Amazon.com reviews - surprisingly the one bad review I read gave me details on a free calculus book written by a professor in U of Wisc which he offers for free to download and apparently the reviewer suggests that that book was of a higher quality. - what Luck!

[oh a spoiler aler!]http://www.math.wisc.edu/~keisler/calc.html[/spoiler!]

Thoughts + regards?

PS: if this is in the wrong section please let me know?

 

[PowerIso]

The reason why so many math majors love Spivak is because it isn't just applications. Most of it is rather theoretical and rigorous. I never used it, I have used Apostol though, so I appreciate books that pushes a student pass their comfort zone.

 

[heldervelez]

I've appreciated the Calculus of Spivak (long time ago). But Apostol is also fine.
I've looked to the 1st chapter of the book you mention on OP, and I liked it.
Its a new way to be explored by teachers.
(but the teachers are biased because they already know 'this way', and pay attention to 'another way' needs time and effort).

I wouldn't be surprised if this new approach proves to be a more easy way to calc.

 

[jbunniii]

Spivak is good preparation for those who intend to major in mathematics or a related field that will require studying real analysis. It is "conventional" in the sense that it works with standard real numbers and uses the usual epsilon/delta arguments for limits.

The free online book you referenced is unconventional because it uses "infinitesimals," which are positive entities that are smaller than any positive real number. These objects, once derided by Berkeley as "ghosts of departed quantities," are in fact a perfectly legitimate construct if you do it correctly (which I assume the Wisconsin author does). Learning calculus this way is not wrong per se, but bear in mind that it will lead you down the path toward "nonstandard analysis," which will prove disadvantageous or at best orthogonal to later study of (standard) analysis as taught in most undergraduate programs.

The "big deal" about Spivak is that it is one of only a handful of calculus books that are fully rigorous and don't gloss over the technicalities. This is important for math majors and others who can't or don't want to accept faith-based mathematics. The other "big deal" about Spivak is that it's more fun to read than the obvious alternatives, such as Apostol or Courant, though I personally have a fondness for the latter.

I haven't read the negative Amazon review in question, but if the reviewer is seriously putting forth a nonstandard/infinitesimal calculus book as a preferred alternative then he is comparing apples to oranges, and I question his agenda. People make a big deal about Spivak because it's a fantastic book and the few who dissent often seem to have an axe to grind (or are not mathematically ready for Spivak).

 

[emyt]

my favourite thing about the spivak book is that the questions are very instructive

 

[heldervelez]

Spivak is good ... It is "conventional" in the sense that it works with standard real numbers and ...
... book you referenced is unconventional because it uses "infinitesimals," which ... are in fact a perfectly legitimate construct if you do it correctly ... Learning calculus this way is not wrong per se, but bear in mind that it will lead you down the path toward "nonstandard analysis," which will prove disadvantageous or at best orthogonal to later study of (standard) analysis as taught in most undergraduate programs.

The "big deal" about Spivak is that it is one of only a ... Spivak because it's a fantastic book ...

I agee with almost your post Mr. jbunniii.
If I were a student now I would choose Spivak (or the appointments of a wonderfull prof I had back in the 70s (prof. Campos Ferreira of IST, Lisbon) that grab the attention of 200 young fellows, smoking cigarrete after cigarrete and the microphone in one hand and the chalk in the other hand. To him we are not there until he asks 'any question?'. To each element in the audience only the prof existed. I left this personnal note as a testimony to his memory).

I've looked at http://en.wikipedia.org/wiki/Hyperreal_number"
and it seems it is self-consistent, and it is more rich than consider only Reals, and I understand the 'discomfort' that Abraham Robinson must felt with 'conventional treatment'.
I've studied 'conventional' analysis and we I am able to 'rephrase' to other 'unconventional view', and I think that the vice-versa could be also possible without rupture. I put in question your saying which will prove disadvantageous. Only with testing we could say which one approach is better TO THE FUTURE. Because the future must be constructed at any moment, even now.
I say again that if I were a teacher I'd have to do a comparison and decide. And possibly try lobbying for adoption.

As an example of evolution, I've recently discovered http://en.wikipedia.org/wiki/Interval_arithmetic" and it is not 'standard' and I will use it in a class of problems that deserved other approaches back in time.

We must be open-minded and adopt new ideas if they are more operational than 'conventional' solution. Dont attach to the past by the simple fact that all books are 'conventional'. What is conventional now was a revolution back in time.

But the position of the students must be precautious and learn well one method (Spivak is exccelent and preferable to Apostol, at least to start with analysis, I have both).

The students now are learning that 'space is expanding' , the Big Bang existed back in time, Dark matter must be around and also Dark energy, and billions of pages say so. But when a new emergent theory (in fact already available (*)) say against those sayings what must we do ? RECONSIDER, analyse, rewrite.
Go to the future. The students of today must be more curious than their teachers, because sometime, in time, they have to reconsider, even if they learned 'this way'.


(*) The politics in use in this forum say to me 'no-go' to such a discussion and even the saying 'hey look there' is forbidden. But it is a strange thinking because PHYSICS is all about change and evolution.

It is the in the very nature of humanity and heritage, the questioning of past knowledge, and try to find new and better ways.