Good supplementary books for a physics and math dual major?

Question

I was wondering what books would be good for helping develop a more sophisticated understanding of concepts in maths for a physics/maths degree. For reference, I'm doing a def-thm-proof style Real Analysis course (I'm a first year), and I find that while there are some books to help me further iron out creases that emerge from lectures (Rudin + Abbot together), I can't really find that for physics. I'm not so interested in books on physics subjects, but rather the methods. I was talking to my RA tutor and he mentioned that a proper treatment of differential forms is important for understanding why physics texts derive equations by "now make this a small change dq resulting in a dF" and it still works.

I was pointed to David Bachman's Geometric Approach to Differential Forms, for a proper treatment of the subject for this hand-wavy "make this dx" shenanigan that arises often in physics, but I was also suggested that going into my second year to study Gelfand and Fomin's book on Variational Calculus. I was just wondering if there are better options out there for these texts, and if there are other books or topics that I should be aware of to prepare for my studies (for me it's difficult to just accept the hand-waving and I'd like to approach these topics with more rigour lest I don't really understand what's going on).

I think closed integrals on surfaces and lines is in a weird state at my university between assumed knowledge and to-be learnt knowledge, so if there are any good multivariable calculus books or the like (I've studied linear algebra too), I'd greatly appreciate any suggestions. Thanks!

Answer 1

As a general book to develop a more "sophisticated" understanding, i would recommend the book "Mathematical Physics" by Robert Geroch. It provides expositions to many different areas of mathematics that are useful for physics in the "right" way (this is subjective of course). It (together with knowledge of multivariable analysis) is also a great "gateway drug" to more advanced texts.

As a book on multivariable analysis and basic differential geometry i would recommend the book "Calculus On Manifolds" by Spivak. It is very economic (only around 135 pages) and contains the necessary multivariable analysis and basic differential geometry. Furthermore the solutions to the problems can be found online which is ideal for self study. The needed prerequisites should also align with your current knowledge level. To quote the preface:

The formal prerequisites include only a term of linear algebra, a nodding acquaintance with the notation of set theory, and a respectable first-year calculus course [...].

As for the book you were recommended ("Geometric Approach to Differential Forms") it seems that the second edition made the book less accessible (and probably not a good recommendation for you). To quote the preface:

The original edition of this book was written to be accessible to sophomorelevel undergraduates, who had only seen one year of a standard calculus sequence. [...] Hence, for the second edition, most of the material on basic topics such as partial derivatives and multiple integrals has been removed, and more advanced applications of differential forms have been added.

So if you want to read this book you should probably get your hands on the first edition and not the second.

As for a calculus of variations book, i personally did not enjoy Gelfand and Fomin at all. The first part of the book "Calculus Of Variations" by Jost is an elementary introduction (the first part relies only on multivariable differential calculus and the Riemann integral) to the calculus of variations as it is used for example in (Lagrangian) classical mechanics.