Physics and Math Double Major with self-study

[Bashkir]

I currently have completed a two-semester course in physics where University Physics by Ronald Reece was used. This two-semester course was your basic introduction: the first semester comprised of Newton's laws and their extension to energy, momentum etc., and the final semester was a light introduction to thermodynamics and then the basic principles of E&M were covered, AC and DC circuits (non-complex analysis), The integral form of Maxwell's equations etc.

In the realm of mathematics, I have not had a college level mathematics course. My understand of calculus is limited to basic differentiation, integration, basic taylor series, and a cursory knowledge of some vector calculus from physics.

I feel like I didn't acquire a stable enough conceptual and computational foundation from my first year in college. Due to personal issues I was forced to withdraw from Uni and now I am taking this next year to self study.

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I am currently working through An introduction to Mechanics by Kleppner and Kolenkow, and after that I plan to work my way through Purcell, or Griffiths. I'm leaning towards Purcell because the quality of problem seems a little better. Unfortunately, the non-SI units are a pain, but I don't want to shell out the $60 dollar difference to pay for a newer addition. After this however, I don't know where to go.

What are some good recommendations for a good introduction to modern physics that is of the same quality/information?

The modern course offered at my university was optics, relativity, then intro to atomic physics and then wavefunctions and basic Schrodinger equation stuff. I don't know if this is the general structure of a modern course or not, but the book doesn't need to follow this pattern.

After finishing the modern course, would it better for me to move to an intermediate text for mechanics and E&M like Morrin and Griffith, or are there better textbooks/would it be advantageous for me to move to Goldstein/Jackson if I am decently mathematically prepared giving that I have a decent understanding of the conceptual grounds.
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For mathematics the situation is a little complicated since I have such a sparse mathematical background and I'm considering doing a mathematics double major.

For calculus I'm considering Introduction to Calculus and Analysis by Courant and John, but I'm not completely sold. I have looked at Spivak and it seems to go a little more in-depth, and Apostol is just too dry for me personally. Are there any better recommendations? Also, I know this will probably be subjective, but should I be considering with getting a rigorous introduction? Will it actually be worth it later when studying higher mathematics? I figure that I will learn all of the nuances eventually, but am unsure of whether to start big, or do simple texts like stewart and just build from the ground up.

After the calculus, I am pretty lost on where to go. I know what mathematics courses I need to take, and this is how I was thinking of doing them.

Calculus (Single,Multi, and vector)
Discrete Mathematics (Unsure if I should really study this or skip and just take a university course later)
Linear Algebra (I have a book by paige swift, but I'm not sure how it holds up and seems lacking in problems)
Differential Equations (ODE, PDE)
Abstract Algebra
Abstract Linear
Analysis (Real, then Complex) Probably going to go with Rudin. I would love to hear some suggestions on the books and the order of intended study. I realize that I will not be able to do any of this studying justice if I did it in just my year off. I plan on consistently studying throughout the rest of my university career as well as a supplement to my courses.

I realize that some of these questions have been frequently asked, but for some of them I have yet to find a satisfactory answer.
I apologize for the more massive than I intended post haha.

 

[atyy]

For EM, a book I like very much is David Dugdale's"Essentials of Electromagnetism". It's slightly easier than Griffiths, but what I like about it is that it starts from Maxwell's equations. So you know the "complete truth" of classical electromagnetism from the start.

For classical mechanics, it's very important to know the Lagrangian, Hamiltonian, and Hamilton-Jacobi formulations. It's also important to know about integrable and non-integrable systems, and action-angle variables for integrable systems. A readable book is Fetter and Walecka's "Theoretical Mechanics of Particles and Continua".

For quantum mechanics, it is again important at some stage to have the "complete postulates". I prefer the stage to be early, so I like Sakurai and Napolitano and Nielsen and Chuang. Unfortunately those don't have too much interpretation in them, so one needs to also read Landau and Lifshitz, or Weinberg. I think it is a bit hard to start with Sakurai, as one should see the triumphs of quantum mechanics in blackbody radiation, atomic spectra etc so that one knows it works. French and Taylor, or Eisberg and Resnick are good books to start.

Quantum mechanics needs linear algebra, which one can learn from Schaum's series.

At some stage, I think it would be nice to read Spivak's "Calculus on Manifolds". The main reason is to see Green's theorem, the divergence theorem and Stokes's theorem as "the same" theorem. Although Spivak's book is supposed to be rigourous, it is so well written that it can also be read non-rigourously.

 

[Bashkir]

Thank you for your detailed response!

How close in mathematical difficult would you say that Griffith's and Fetter and Walecka are?

 

[atyy]

I think Fetter and Walecka's mechanics text is at about the same mathematical level as the Griffiths EM book.

Regarding Purcell: I think one can use any EM book that one wants, since the material is about the same, and different people will have different tastes. However, there are two things in Purcell I have rarely seen elsewhere: a "physical" derivation of the Lamor formula, and a simple example of how B fields and E fields transform between frames.

Schroeder liked these examples, but apparently his students didn't like Purcell, so he tried to make a simpler presentation of the Purcell example, which I like: http://physics.weber.edu/schroeder/mrr/mrr.html.

 

[Bashkir]

Thank you very much. That looks perfect.