Apostol vs Spivak Calculus

[lewelma]

DS is very interested in theoretical maths, which is why I'm looking at both of these books.  He has already completed Calc 1 and 2 using a combination of AoPS Calculus and Anton Calculus -- AoPS being theoretical and Anton being practical.  He feels like he has forgotten quite a bit of it given that it has been a year since he finished the books.  Since then he has taken an Analysis course and a linear algebra course at the local university.  He is interested in studying multivariate calculus to a degree that he could take a placement exam at whatever school he attends and place out of all 3 calculus courses (we are in NZ so APs are not really an option, and I am thinking placement exams at elite universities, so I expect they will be hard). 

 

I have done some reading and it looks like only Apostol covers multivariate material, but it is less theoretical than Spivak.  I'm trying to decide if he should buzz through the Spivak Calc for review/enrichment and then move on to the second volume of Apostol.  Does that make sense? Or should I get both volumes of Apostol and skip Spivak?  I understand that there are only solutions for Spivak.  I'm not sure if that will be a problem for ds with Apostol as ds is pretty good with maths. ;)

 

Thanks,

Ruth in NZ

[Kathy in Richmond]

I've only used Apostol, so I can't compare the two books for you. I can't imagine needing a more theoretical approach than Apostol, though. :001_smile: Pick one and delve into it...no rushing if you want to get the most out of it. I had my kid pick sections from Apostol for senior year, and it's definitely a good prep for the type of math you'd encounter at a top tier university.

 

To be honest, I'm not sure that those books would help him a lot with placement exams. Most of the colleges your son is considering offer different levels of calc and multivariable math. Spivak and/or Apostol would be one of their honors levels, but the typical placement exams test at the regular (albeit not easy peasy!) level.

 

You can look up a final exam for 18.02 (multivariable calc) at MIT online. For example, I found this one on MIT OCW.  Lots of problem solving, but no proofs that I can see. Spivak & Apostol are more proof oriented texts.

 

My kids used Marsden & Tromba's vector calculus text for multivariable calc at home (before Apostol). It's rigorous (no hand waving), ties MV into physics and some real analysis, and has tough problem sets. When ds went off to MIT, he reviewed from that book before taking and passing the 18.02 placement exam. So it worked fine for that level.

[lewelma]

Thanks Kathy!

 

I've looked up the Marsden book and tried to compare it to other non-apostol style books.  Marsden seems to be considered more rigorous than most of the others, but also harder to read.  It is so hard to figure out how to interpret reviews unless you know the reviewer's skill level, which obviously I don't.  DS has taken linear algebra, so the chapter you linked to, he said "this is easy, I know all this stuff." So perhaps reviewers who struggled to read the book weren't as versed in linear algebra as ds.  How did your kids find Marsden?

 

DS also wanted me to ask you if he could just do Apostol, rather than Marsden and just dabbling in Apostol.  He has a full year to do multivariate calc rather than just a term, although it is a pretty full year! Is Apostol better done after a more standard approach?  Does Apostol take too long to do?  Why did your dc only do selected topics? DS used Anton last year after he did AoPS calc, and he told me that likes Anton.  He wanted to know if he could use Anton for vector calc basics and then do Apostol?  Or would this not serve him well for the placement exam?  Kind of like low level and high level, but the exam is mid level?

 

I know that was a lot of questions, but ds is leaving tomorrow for Rio!  So he just read your comments, some comments on line, and then threw a ton of questions at me!   :001_smile:  I think his mind is elsewhere! Its a bit like  :willy_nilly: over here today.

 

 

[Kathy in Richmond]

Thanks Kathy!

 

I've looked up the Marsden book and tried to compare it to other non-apostol style books.  Marsden seems to be considered more rigorous than most of the others, but also harder to read.  It is so hard to figure out how to interpret reviews unless you know the reviewer's skill level, which obviously I don't.  DS has taken linear algebra, so the chapter you linked to, he said "this is easy, I know all this stuff." So perhaps reviewers who struggled to read the book weren't as versed in linear algebra as ds.  How did your kids find Marsden?

 

We found Marsden easier to read than Apostol here, though we enjoyed both books. My kids found it to be the perfect level between AP BC calculus and Apostol & other higher level maths. We also like the applied physics flavor of Marsden; Apostol is more pure mathematics in flavor.

 

Here's a review from the AoPS boards for you that might help. I originally chose the book based on another review I'd read on AoPS way back when ds was in high school. It also helped that I'd met Jerry Marsden in grad school (he collaborated with my PhD advisor at CMU) and thought he was brilliant. Later dd and I "met" Tromba while touring MIT dorms...he was there with his dd, too :-)

 

The book does start out with a definition of vectors and their basic properties. Easy linear stuff? Yeah. But it goes on to much more, weaving strands of linear algebra and real analysis into its explanations of the calculus of several variables. For example, flipping open to a random page, I see that it really delves into the change of variables formula for double integrals & explains the theoretical math behind it (more so than Anton or Stewart do).

 

 Here's a link to the full table of contents.That's the edition I like best, btw. When dd took my volume with her to college, I purchased the 4th edition & didn't like it as much. It seemed to have lost a lot of the nice historical sections linking the mathematics to physics.

 

DS also wanted me to ask you if he could just do Apostol, rather than Marsden and just dabbling in Apostol.  He has a full year to do multivariate calc rather than just a term, although it is a pretty full year! Is Apostol better done after a more standard approach?  Does Apostol take too long to do?  Why did your dc only do selected topics? DS used Anton last year after he did AoPS calc, and he told me that likes Anton.  He wanted to know if he could use Anton for vector calc basics and then do Apostol?  Or would this not serve him well for the placement exam?  Kind of like low level and high level, but the exam is mid level?

 

Yes, I think that Apostol is best used after an intial exposure via Anton or Marden/Tromba. Your ds could do both Anton and Apostol and fill out a year of study easily. I personally learned the basics of MVC from Thomas (old 70's edition) & then Apostol, followed by a year of TA'ing out of Apostol. That really helped me connect all the dots! If he does both books as he suggested and works through an old placement exam for practice, I think he'd be in fine shape.

 

We only chose selected topics in Apostol because it takes LOTS of time to go through both volumes and work the exercises well! And we were just too busy, too. During ds's senior year, he studied from Apostol, took WOOT and Olympiad Geometry, prepared for & wrote the USAMO, took part in other math competitions, and had lots of other subjects and college applications, too. Busy times!

 

I know that was a lot of questions, but ds is leaving tomorrow for Rio!  So he just read your comments, some comments on line, and then threw a ton of questions at me!   :001_smile:  I think his mind is elsewhere! Its a bit like  :willy_nilly: over here today.

 

Good luck to your son in Rio! I'll be rooting for him and his team!!

 

Edited July 9, 2017 by Kathy in Richmond