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I am currently a first year student studying Theoretical Physics. This summer I had a mind to try and become well versed on the topics of multivariable/single variable calculus and linear algebra, as I consider them to be a necessary foundation on the road to building an advanced mathematical knowledge.

Currently I have taken a full year of linear algebra and a module in single variable and advanced calculus. I was recommended the books Advanced Calculus (Loomis and Sternberg) and Steven Romans Linear Algebra, however I feel the latter may be a bit out of my reach at the moment.

I would greatly appreciate some advice from someone more experienced who also once stood where I am currently. Thank you in advance!

benmcgloin
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    I have no idea why someone would think those two books should be recommended, as they're well out of the reach of a large majority of pure math undergraduates (even 4th year students), and they aren't particularly suitable for graduate students in physics either. I suggest before getting involved in esoteric pure math topics (and when you do, basic general topology and manifolds level differential geometry and functional analysis would be better areas to focus on) that you make sure you know very well things covered (continued) – Dave L. Renfro Mar 25 '21 at 14:33
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    in books such as Marsden/Tromba's Vector Calculus (probably you have this background already) and Mary L. Boas' Mathematical Methods in the Physical Sciences and Marsden/Hoffman's Basic Complex Analysis (see last few chapters/sections for why I picked this). At some later time, a good entry point to functional analysis for a physics student is Erwin Kreyszig's Introductory Functional Analysis With Applications. – Dave L. Renfro Mar 25 '21 at 14:42
  • I have heard bad reviews of Marsden/Tromba's, unimaginative and the like. Would Hubbard and Hubbard's get your recommendation? It has many excellent reviews from people at my college and seems to cover much of the same material as Marsden. Also, thank you for taking the time to respond, you have already been a great help! – benmcgloin Mar 25 '21 at 14:52
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    Yes, I think Hubbard/Hubbard would be fine. I don't have a copy of that book, but it seems fine from what I can tell. A bit more advanced is Charles Henry Edwards' Advanced Calculus of Several Variables, but probably try this only if you know multivariable/vector calculus well at the elementary level. Incidentally, a bit more theoretical but still close to the same level is Wendell Fleming's Functions of Several Variables, but I think Edwards' book is better suited for a physics student than Fleming's book. – Dave L. Renfro Mar 25 '21 at 15:04
  • I don't know how you can go though a Bachelor's in Physics without linear algebra. You need the complete mathematical tool set. I imagine any introductory books on linear algebra, real analysis (and multivariate), and ordinary and partial differential equations would suffice. Interesting to physics are hyperbolic PDEs. – superAnnoyingUser Mar 26 '21 at 15:35
  • Lars Ahlfors' book is particularly good for complex analysis of one variable and Vladimir Arnold's for ODEs. The Berkley course in thermodynamics treats the subject particularly well, etc. But man, the scope of your question. You need any book on Riemannian geometry then who knows. At some point you will be ready for Springer's Mathematical Physics books and the Series on Knots and Everything. – superAnnoyingUser Mar 26 '21 at 15:45
  • Hi Ben. I don't think a book like Loomis and Sternberg is crazy for someone who is very interested both in math and in physics, and is highly proficient in theoretical single-variable calculus (such as in Spivak's book). That being said, I agree with Dave Renfro that it will be beyond the ability of the majority of first-years. Assuming that you are interested more in practical mastery and less in "epsilonics", I'd recommend you have a look at "An Analytical Calculus" by Maxwell, particularly Volumes III and IV. Despite being in four volumes, it's actually quite concise, it has good problems.. – Anonymous Mar 29 '21 at 23:01
  • and it's free to download from archive.org. The first two volumes correspond to what used to be done at A-level in England, the third to Cambridge entrance scholarship exams, and the fourth to the first year of university. It doesn't cover vector calculus (div, grad, curl), but it does have a lot on diffreential equations. I also like the book by C.H. Edwards recommended by Dave Renfro, and it has an introductory review of basic linear algebra. For more advanced linear algebra, I'm not in a good position to recommend a book for you, as the books I know would be better suited to... – Anonymous Mar 29 '21 at 23:06
  • math specialists. – Anonymous Mar 29 '21 at 23:08
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    Sorry, let me correct a mistake. Maxwell's book is not free to download from archive.org, but it can be read online for free if you have an account at archive.org. – Anonymous Mar 30 '21 at 00:28
  • Thank you for taking the time to respond, I’ll look into Maxwell’s book. I would say I am more interested in the mathematical side of my course, I am actually currently deciding whether I will switch to pure mathematics next year. I have a copy of Loomis at hand and also ordered some of the books recommended above. – benmcgloin Mar 30 '21 at 10:02
  • All right. The books recommended above are mostly suitable for people trying to transition to rigorous math through the medium of multivariable calculus. This makes sense if you want to be introduced to pure math but also want to get on with the math you need for physics. If you want to take a step back and look at the single-variable material you may already have studied, but not rigorously enough, from a more theoretical standpoint, Burkill's concise First Course in Mathematical Analysis is great. The (linear and abstract) algebra book I'd recommend for pure math is Algebra by Artin. – Anonymous Mar 30 '21 at 14:35

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