Calculus 2 Apostol

[Liora Putcha]

Whichone of those is the best for a person interested in pure mathematics and who wants to have a deep understanding of calculus? Apostol or Spivak? Could you guys tell me some differences between the approaches of them? What about the exercises? I would like to be challenged, but in a constructive way.

While both books have complete proofs and a good emphasis on theory, Spivak's book is better as an introduction to rigorous math because many of its problems are more difficult and theoretically oriented than Apostol's. (I assume you mean his Calculus). Spivak's book also has a solution manual, which is very useful when you're studying on your own.

On the other hand, Apostol actually covers more material, even just within Volume 1. Volume 2 of Apostol is actually one of the best introductions to multivariable calculus. Apostol's book also has a greater variety of exercises involving applications of calculus to physics.

Both books give you an introduction to calculus, being: set and number theory, some induction, series and notation. Big difference can be the order in which topics are introduced, for Apostol integration comes first due to historical reasons, while for Spivak derivatives comes first.

For me, Apostol provides an interesting framework to learn about how calculus' ideas evolved historically, as well volume 1 introduces lineal algebra, providing a more gentle progressions towards multivariable calculus and differential equations in volume 2. On the hand, Spivak goes deeper into proofs with rigour, which is essential for every mathematician or even physicist. Therefore, I decided to use Apostol as my "guide" book and Spivak to delve deeper into concepts in which I'm interested.

There are several reasons why one might consider switching from Tom Apostol to James Stewart for calculus textbooks. Some possible reasons include the level of difficulty, the teaching style, or the specific topics covered in each textbook. Ultimately, the decision to switch will depend on individual preferences and needs.

The main differences between Tom Apostol and James Stewart's calculus textbooks lie in their approach and level of difficulty. Apostol's textbook is known for its rigorous and advanced coverage of calculus, while Stewart's textbook is more widely used and offers a more accessible and user-friendly presentation of the material.

Tom Apostol's textbook is generally considered to be more challenging and geared towards advanced students, while James Stewart's textbook is designed to be more approachable and suitable for a wider range of students. However, both textbooks cover the same core concepts and principles of calculus.

This ultimately depends on the individual's learning style and preferences. If someone prefers a more rigorous and challenging approach, Tom Apostol's textbook may be a better choice. However, for those who prefer a more user-friendly and intuitive presentation, James Stewart's textbook may be a better fit.

It is unlikely that switching from Tom Apostol to James Stewart will significantly impact one's understanding of calculus. Both textbooks cover the same core concepts and principles, and the decision to switch will ultimately depend on individual preferences and needs. It is important to choose a textbook that aligns with one's learning style and goals.

Apostol's Calculus, also known as Calculus: Volume 1 and 2, is a widely used textbook written by mathematician Tom M. Apostol. It covers topics in both single and multivariable calculus and is known for its rigorous and thorough approach to the subject. It is important because it has been used as a standard textbook for many calculus courses and has influenced the teaching and understanding of calculus worldwide.

The books that have been published after Apostol's Calculus continue to build upon the foundations and principles established in Apostol's work. They often cover more advanced topics in calculus and offer new perspectives and approaches to the subject. These books are important for students and researchers looking to deepen their understanding and apply calculus to real-world problems.

Some popular books that are often recommended to read after Apostol's Calculus include "Calculus" by Michael Spivak, "Advanced Calculus: A Differential Forms Approach" by Harold M. Edwards, and "Real and Complex Analysis" by Walter Rudin. These books delve deeper into the concepts and applications of calculus and provide a more challenging and enriching learning experience.

It is not recommended to use books after Apostol's Calculus as standalone textbooks, as they often assume a strong understanding of the concepts covered in Apostol's work. These books are better suited as supplemental reading material for students who have already studied Apostol's Calculus or as reference books for more advanced topics in calculus.

Yes, there are many online resources available for books after Apostol's Calculus, including lecture notes, practice problems, and video tutorials. Some popular websites for these resources include Khan Academy, MIT OpenCourseWare, and Coursera. These resources can be helpful for students looking to supplement their understanding of calculus or for self-study purposes.

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. ;)

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.

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.

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 :-)

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