# Introductory textbooks in mathematics [duplicate]

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Some time ago I started self studying physics and I did quite well, despite my limited knowledge of maths (some basic algebra and geometry). Now I would like to understand calculus, but I don't know what books I should use. Clearly, first I need to revisit the basic and intermediate notions of algebra. For this task I tried using this book:

https://www.amazon.com/Ron-Larson-Intermediate-Algebra-fifth/dp/B008UBLER4

but I found it to be somehow simplistic (problems are too easy, the theorems are not demonstrated, etc.)

What textbooks do you think are appropriate for someone with very basic knowledge of elementary algebra who wishes to self study calculus for use in physics?

## marked as duplicate by rschwieb, Ethan Bolker, projectilemotion, Ken Duna, HK LeeApr 1 '17 at 18:37

• Perhaps a precalculus text? – MPW Mar 30 '17 at 16:23
• There are dozens of recommendations for calculus texts, some even for just physics. You should go read those posts first before making another post that asks all over again. – rschwieb Mar 30 '17 at 16:29

I give a general answer. It is very nice to read the linear algebra first. I myself used the book Linear Algebra which was very helpful and is well-written. You can use other textbooks in linear algebra by searching them in Google. For calculus you can use the book Calculus by James Stewart and for differential equations there is the book Ordinary differential equations by Boyce and Partial differential equations that consists of completely solved questions and is very nice.

If you need to know more about mathematical subjects and methods in physics, you can use the books Mathematical Methods for Physicists by Weber and Arfken and Mathematical Methods for Physics and Engineering which both have some elementary explanations about group theory in Abstract Algebra.

For further readings I advice books on Functional Analysis since in physics you are working with infinite dimensional function spaces (like the Fourier space) and Hilbert or Banach spaces and operations between them, so I recommend you to read An introduction to Hilbert space and Real Analysis: Modern Techniques and Their Applications which consists some abstract concepts and notions. Also An epsilon of room real analysis by Terrence Tao which is very nice and well-written.

EDIT: Another recommendation is to search any topic you see when you're studying in Google Books. Also for differential geometry you can use the book Elementary differential geometry by Andrew Pressley.

Good luck!

• Ted Shrifin also has a multivariable calculus book which covers the linear algebra, topology and language of differential forms + tensors needed for studying physics at the university level. amazon.com/Multivariable-Mathematics-Algebra-Calculus-Manifolds/… (this is for the OP). Great suggestions by the way. – Faraad Armwood Mar 30 '17 at 16:44
• @FaraadArmwood I forgot to add geometry topics which you remind me of. – MohammadSh Mar 30 '17 at 16:55

Contrary to belief, a rigorous calculus text is right up your alley! Books such as Apostol I,II and Spivak's calculus start of by developing your understanding of the real numbers which will require you to work on your algebra skills. To understand physics outside of the university level (a first year's course) you'll need much more than calculus.

A really good understanding of multivariable calculus will get you through most undergraduate physics courses. Therefore a book which takes you through the whole calculus sequence is more desirable since you don't want to waste time trying to adjust to a new writer's style + notation. Any of these two books mentioned above provides a nice 2 for 1 in your case since they'll make you improve your algebra skills before the calculus starts.

Personally, I didn't learn from the books I've suggested, but I wish I would have. I've noticed that although I know how the ideas in calculus work (since I do a lot of differential geometry, etc), I am sometimes fuzzy on the analysis to rigorously prove arguments that I know to be true. As mathematicians, we care very much about details, especially when the big picture is clear and we want to smoothen out an argument. However, a physicists may not be to concerened about these things, so I also suggest reading other posts related to this one, as other users have suggested.

$\textbf{Comment}$:I believe I am okay in saying this, check abebooks for copies of the texts I mentioned above. They sell international copies which are much cheaper that those you'll see on amazon. I hope this helped.

• Maybe it depends on the institution, but at my university (UChicago) multivariate calculus is enough to get you through the first year sequence. Linear Algebra, Complex Analysis, and basic Field Theory is essential for understanding quantum mechanics at the level taught to third years. Let alone fourth year when group theory becomes necessary to understand gauge theory. I can't speak for other disciplines of physics having only taken the QM courses, but I would imagine topology is necessary in any course on GR too. Plus a serious knowledge of differential equations seems an obvious prereq. – Stella Biderman Mar 30 '17 at 17:03
• Yes, UChicago and schools alike are definitely and exception. If you want to get some perspective on how calculus is taught generally, we can just look at texts such as rogawski, penny, etc and we quickly see that not much is done to show connections between calculus, physics, linear algebra etc. These texts were designed to show people for the most part, how to do things, versus why things are done. – Faraad Armwood Mar 30 '17 at 22:38

I would recommend one of two approaches.

The first would be to try reading a calculus text with extensive review sections for algebra. An example of this would be Calculus by Marsden and Weinstein. This is a reasonably good textbook for non-rigorous calculus. By this I mean that, like most calculus textbooks used in North America, but unlike Apostol and Spivak's calculus books, it doesn't focus particularly on theory and doesn't give full proofs of all theorems. The book reviews the necessary material from algebra and trigonometry, and even has quizzes that tell you whether you need to read certain review sections, or whether you even need to go back and read a precalculus book.

The other would be to read a precalculus book. One of the best books of this kind is Basic Mathematics by Lang. The same author has a calculus book, A First Course in Calculus, that I would recommended highly as a book intermediate in difficulty between Apostol or Spivak on the one hand and non-rigorous texts like Marsden/Weinstein or Stewart on the other. Lang's book is quite concise in comparison.

If you're in a hurry to learn just enough calculus for physics, one possibility would be Quick Calculus by Kleppner and Ramsey.