What Books to Read After Apostol's Calculus?

[Akorys]

Hello everyone!

This is my first post on the forums so I hope I'm not posting this in the wrong section.

I have been working through Apostol's Calculus Vol. 1 and so far I've been enjoying it a lot. I'm about to finish the 5th chapter, and while I would ordinarily wait until I was almost done with the book, I find that it would be easiest for me to get one or two books earlier than I had planned. I am definitely planning to buy the second volume of Apostol, but I would appreciate any advice on what the second book could be.

I understand that linear algebra would be a beneficial topic to cover, and have considered Axler's "Linear Algebra Done Right". However, I see that Apostol covers linear algebra in his calculus books and I'm afraid that most of the material would be repeated. I also looked at Artin's "Algebra" and I would love to introduce myself to the topic of abstract algebra.

If my experience is relevant, I am self studying this material and have only had access to math beyond high school through this book.

Thank you!

 

[Akorys]

I'm sorry if my questions were not clear. I'll try to restate them more briefly;

1. Is "Linear Algebra Done Right" by Axler a good book after having read Apostol's Calculus?
2. Is "Algebra" by Artin a good book after Apostol?
3. Is there something different I should focus on after Apostol?

 

[slider142]

I'm sorry if my questions were not clear. I'll try to restate them more briefly;

1. Is "Linear Algebra Done Right" by Axler a good book after having read Apostol's Calculus?
2. Is "Algebra" by Artin a good book after Apostol?
3. Is there something different I should focus on after Apostol?

Those are both excellent texts to follow Apostol, as you will need a good grounding in algebra in order to make the most of the study of differential equations and differential geometry. You may want to go ahead and study Spivak's "Calculus on Manifolds" on the side as well, so that you get a good introduction to the modern language of exterior calculus. Just for completion, you should also study a text that focuses on complex analysis, such as Ahlfors, and Needham's brilliant "Visual Complex Analysis" should complete a basic introduction. If you want to focus on any other topics besides plain analysis, you can branch off from here as well, as they will be mostly independent in basis. Ie., statistics, topology, combinatorics, etc.

 

[Akorys]

Thank you for the advice! I'll keep that in mind for the future. Do you think it would be best to first study linear algebra through Axler before delving into Artin's "Algebra", or is the order not too important?

 

[qspeechc]

Yes, those are both good books you can read next. Artin also covers linear algebra, although Axler is often used as a supplement to Artin. Vol.2 of Apostol does indeed cover quite a bit of linear algebra, but the books by Axler and Artin are sufficiently different that you should still study them.

If you're asking what topics you can/should study after Apostol, then you have a variety to choose from. Apostol is basically and introduction to Analysis. The two books you named deal with Algebra, and are perfectly suitable to study after Apostol. You will have seen some linear algebra in vol.2 of Apostol, so you will have some idea of algebra and if you'd like to study it further. Axler treats linear algebra very differently from Apostol, and gives very insightful perspective. Artin covers a lot of other algebra as well and is an excellent book.

You can also carry on with your study of Analysis if you want to: you can either move on to Spivak's Calculus on Manifolds, which develops the ideas in Apostol vol.2, and develops towards differential geometry. Or you could move on to actual Real Analysis, which is basically calculus done in a more sophisticated and advanced way, and I would suggest Apostol's Mathematical Analysis if you enjoy his calculus book,or Pugh's Real Mathematical Analysis. If you feel up for a challenge, you could even make a start on the more topological side of Analysis, which a book like Simmons' Introduction to Topology and Modern Analysis covers. Whichever book you choose, you should be able to start studying topology after that.

Or you could even start studying Complex Analysis; which is exactly what you expect it to be: basically, calculus with complex numbers (cool, right?). There are a very large number of good books on this topic. I learned from vol.1 of Conway's Functions of One Complex Variables, which is fairly good, but not great. Many people, including Mathwonk, recommend Lang's Complex Analysis, but I am not familiar with it.

You can also go into another area of Analysis called Differential Equations. Apostol does cover them a bit, but these books will go into more detail. I know the book Ordinary Differential Equations by Tenenbaum and Pollard has a very high reputation. I learned from Coddington's Introduction to Ordinary Differential Equations and I think it is very good (don't confuse it with his other book, Theory of Ordinary Differential Equations, which is more advanced). After the Ordinary ones, you can go into Partial Differential Equations, but sadly I cannot help you there. I have yet to find a book in that field that I like as an introduction.

Then there are less conventional topics, like Number Theory, for which I suggest An Introduction to the Theory of Numbers, by Niven, Zuckerman and Montgomery. I believe Apostol has even written an introduction to Number Theory as well, but I am not familiar with it. Or you could try some even more unconventional reading, like elementary Geometry, for which I can recommend Moise's Elementary Geometry from and Advanced Standpoint, which deals with school geometry essentially (but which is rarely taught in schools, in fact) with a bit more sophisticated techniques, or you could look at Roe's Elementary Geometry, which also starts with school geometry, but ends with stuff a bit more modern. It is also possible to start on basic Differential Geometry with a book like O'Neill's Elementary Differential Geometry.

Basically, there are a lot of directions you can go in. More analysis, or algebra, or geometry or number theory. The choice is entirely up to you. I'm sure others will add their own suggestions. Have fun.

 

[Akorys]

Thank you both for all the advice, I really appreciate the help!