Rigorous math and physics textbooks

[crat0z]

Hello, physics forums. As an introduction to the community, I'm 15 years old and live in northwestern Ontario. I've recently became very interested in physics, but I've always excelled in math. I've looked into some textbooks, particularly Apostol's I and II, along with Spivak to bridge the two books. As a prerequisite to reading those three, I've also ordered Precalculus by Barnett. For after Apostol, I have bought Borelli and Coleman's Differential Equations. On the physics side of textbooks, I've ordered University Physics by Young & Freedman, but I'm a little confused on what to read after these books.

I read in a thread about Artin's Algebra, and that it covers abstract and linear algebra. With the linear algebra in Apostol's books, I think I should be able to read Artin. I've heard good things about Rudin, but I'm unsure if I should buy Real and Complex Analysis if it is too complicated, especially if I will need knowledge of complex variables for electromagnetism.

Any help is appreciated, I'm a very motivated student for this type of stuff, and I'm willing to work through the most rigorous books in order to understand the mathematical principles behind physics.

 

[Snicker]

You are going too fast. If you're reviewing precalc, there is no reason to even THINK about Big Rudin.

Apostol is okay, but it's quality hardly justifies its price.

It's important to read solid books on algebra so that, if you were to ever crack open some math olympiad book, you wouldn't feel too behind. Algebra by Gelfand is considered an ideal starting point for the young mathematician. Euler's Elements of Algebra is also an amazing read (and not, as one may suspect, archaic).

For calculus, there are several good books. Calculus Made Easy by Thompson is absolutely wonderful (albeit hardly rigorous). Euler himself wrote three calculus textbooks: Foundations of Differential Calculus, Foundations of Integral Calculus, and Introduction to the Analysis of the Infinite. For a (fairly) rigorous treatment, I suggest Elementary Real and Complex Analysis by Shilov. Don't be fooled by its title, I believe that the book was written as an introduction to calculus.

All of Euler's books that I listed can be found for free at http://www.17centurymaths.com/. Calculus Made Easy can be found for free at http://www.gutenberg.org/ebooks/33283 . Gelfand's Algebra has a list price of $32.95, and Shilov's book has a list price of $22.95.

 

[Number Nine]

but I'm unsure if I should buy Real and Complex Analysis if it is too complicated

Rudin wrote two analysis texts: Introduction to Mathematical Analysis and Real and Complex Analysis. The latter is most definitely not an introductory text and you are nowhere near ready for it, and the former, in my opinion, is just not very good. Actually completing all of the exercises in Apostol and Spivak (this seems redundant; I'd recommend Spivak over Apostol) will give you some familiarity with the basics of analysis and proof-writing, so you won't need a completely introductory treatment. There are a few different analysis texts at the appropriate level; one that I'm fond of is Shilov's Introduction to Real and Complex Analysis.

For algebra, you can't really do better than Artin.

 

[crat0z]

Rudin wrote two analysis texts: Introduction to Mathematical Analysis and Real and Complex Analysis. The latter is most definitely not an introductory text and you are nowhere near ready for it, and the former, in my opinion, is just not very good. Actually completing all of the exercises in Apostol and Spivak (this seems redundant; I'd recommend Spivak over Apostol) will give you some familiarity with the basics of analysis and proof-writing, so you won't need a completely introductory treatment. There are a few different analysis texts at the appropriate level; one that I'm fond of is Shilov's Introduction to Real and Complex Analysis.

For algebra, you can't really do better than Artin.

Thank you for the reply, I'm a very fast learner, so I will definitely pick up Shilov's book sometime in the next few months. If you think I should just skip Apostol, what would you recommend for multivariable calculus?

You are going too fast. If you're reviewing precalc, there is no reason to even THINK about Big Rudin.

Apostol is okay, but it's quality hardly justifies its price.

It's important to read solid books on algebra so that, if you were to ever crack open some math olympiad book, you wouldn't feel too behind. Algebra by Gelfand is considered an ideal starting point for the young mathematician. Euler's Elements of Algebra is also an amazing read (and not, as one may suspect, archaic).

For calculus, there are several good books. Calculus Made Easy by Thompson is absolutely wonderful (albeit hardly rigorous). Euler himself wrote three calculus textbooks: Foundations of Differential Calculus, Foundations of Integral Calculus, and Introduction to the Analysis of the Infinite. For a (fairly) rigorous treatment, I suggest Elementary Real and Complex Analysis by Shilov. Don't be fooled by its title, I believe that the book was written as an introduction to calculus.

All of Euler's books that I listed can be found for free at http://www.17centurymaths.com/. Calculus Made Easy can be found for free at http://www.gutenberg.org/ebooks/33283 . Gelfand's Algebra has a list price of $32.95, and Shilov's book has a list price of $22.95.

I left out a few details, and it probably explains why there is some disbelief towards me being able to cover the books I listed above. A year ago, I read through a lot about trigonometry, algebra and calculus (didn't necessarily complete questions), and focused on much of the concepts. I watched through many videos on these fields in math through Khan Academy (whatever that is worth), and I've taken a peak into Apostol I, and I think it would be perfect for me.

EDIT: Money isn't an issue, I come from a somewhat wealthy family, and the new copies of Apostol I bought from abebooks were around $60 in total.

 

[Fredrik]

You need to understand complex numbers, but you won't really need complex analysis until you're at the graduate level (at least). A book on complex analysis will teach you e.g. how to integrate functions along curves in the complex plane, and how to use that knowledge to prove theorems like the fundamental theorem of algebra (every polynomial has at least one root).

Linear algebra is very useful, for special relativity and quantum mechanics in particular. Abstract algebra is less useful. I don't think a physics student will need a whole book on the subject, but it's certainly useful to understand the definitions of the most important terms, e.g. field, vector space, homomorphism, isomorphism, etc.