Pickup and review textbook / material?

[Henry T]

I'm currently studying from a rented Stewarts Multivariable Calculus: Concepts and Contexts however for single varibale calculus I used a lot of different sources for study Kahn, Mooculus and Pauls. I've been self studying calculus for for about a year and a half now. In regards to moving to other mathematics topics soon I've been looking into purchasing a calculus textbook (both single and multivariable) that I can pick up and review from pretty quickly incase I forget a theorem, definition, etc. Stewarts Calculus seems great and I am familiar with the "feel" of the textbook however is there something out there better suited for review or autodidacts?

 

[DevacDave]

I can recommend G.M. Fichtenholz "Differential and Integral Calculus", volumes 1 and 2 should cover all of the material you talk about. It's fairly unheard of outside of the post-Soviet countries, but after being exposed to both sides (USA/UK and Continental Europe/Russia) I am recommending it to everyone. It is very light on dry theorems and puts emphasis on technique an walking you through the problems, not unlike a tutor or good TA. Complemented with something like Rudin or similar rigorous book it is the fullest to my knowledge course for both students and autodidacts.

 

[Henry T]

I can recommend G.M. Fichtenholz "Differential and Integral Calculus", volumes 1 and 2 should cover all of the material you talk about. It's fairly unheard of outside of the post-Soviet countries, but after being exposed to both sides (USA/UK and Continental Europe/Russia) I am recommending it to everyone. It is very light on dry theorems and puts emphasis on technique an walking you through the problems, not unlike a tutor or good TA. Complemented with something like Rudin or similar rigorous book it is the fullest to my knowledge course for both students and autodidacts.

Thanks for the guidance! I'll look into these! Is Stewart a horrid textbook though? I've heard that quite often but I don't find it terrible. I'm planning to be a physics major and the textbook seems adequate for a high school level.

 

[DevacDave]

Is Stewart a horrid textbook though?

To be frank with you, I don't recall having any strong feelings toward it, only that I have had it borrowed from a library for a while. It likely means that it was rather bland and mediocre or I just found something much better for my taste.

After thinking a bit more, I can add a recommendation though: A.I. Kostrikin "Linear Algebra" comes in two variants. One is a single book geared toward physicists and another is a three volume set that is pretty much my favourite set on the subject. Plus there is a very big problems set for it, something that I have never managed to find in USA (student exchange). I'll admit, it may be a bit above high school level but a great thing to have in mind after getting some more preparation. Linear algebra is pretty much essential in modern physics and applied maths, so it would be wise to spend some time on it.

In general, I would try to look for translated Russian books (I can give you more starting points if you would like that) and maybe try to learn bare-bones basics of Russian language anyway. It's by far the most optional step, but most people in USA/UK don't even have a clue about some of the gems that were never translated or popularised. Same thing with German, but to a lesser extent. Give me some more ideas on what you would like and I can find you some more positions, if you are multilingual feel free to add that as well.

Sorry for this 'west does not know how to write' vibe, entirely not what I want to convey. But most of my favourite undergrad-level books are Russian, German and Polish, something that I don't hide. Graduate level is by far dominated by western publishers as far as my favourite books go.

 

[Henry T]

To be frank with you, I don't recall having any strong feelings toward it, only that I have had it borrowed from a library for a while. It likely means that it was rather bland and mediocre or I just found something much better for my taste.

After thinking a bit more, I can add a recommendation though: A.I. Kostrikin "Linear Algebra" comes in two variants. One is a single book geared toward physicists and another is a three volume set that is pretty much my favourite set on the subject. Plus there is a very big problems set for it, something that I have never managed to find in USA (student exchange). I'll admit, it may be a bit above high school level but a great thing to have in mind after getting some more preparation. Linear algebra is pretty much essential in modern physics and applied maths, so it would be wise to spend some time on it.

In general, I would try to look for translated Russian books (I can give you more starting points if you would like that) and maybe try to learn bare-bones basics of Russian language anyway. It's by far the most optional step, but most people in USA/UK don't even have a clue about some of the gems that were never translated or popularised. Same thing with German, but to a lesser extent. Give me some more ideas on what you would like and I can find you some more positions, if you are multilingual feel free to add that as well.

Sorry for this 'west does not know how to write' vibe, entirely not what I want to convey. But most of my favourite undergrad-level books are Russian, German and Polish, something that I don't hide. Graduate level is by far dominated by western publishers as far as my favourite books go.

Thanks for all your help, this is fantastic. I'm taking Linear Algebra next so I'll definitely look at Kostrikin before I make a decision. I don't think I'll be learning Russian anytime soon (no harsh feeling I'm just horrific with language my brain isn't built for it). I do agree that the 'west does not know how to write" a lot of textbooks I've looked over have controversy on how good they are. Thanks again for all your help!

 

[DevacDave]

Thanks for all your help, this is fantastic. I'm taking Linear Algebra next so I'll definitely look at Kostrikin before I make a decision. I don't think I'll be learning Russian anytime soon (no harsh feeling I'm just horrific with language my brain isn't built for it). I do agree that the 'west does not know how to write" a lot of textbooks I've looked over have controversy on how good they are. Thanks again for all your help!

No problem, especially in regard to languages. I honestly know only enough to work through some of the books that never got translated and would not dare to start a conversation in Russian ;). But here are some of the authors who wrote (at least in my opinion) mostly gems:

Vladimir I. Arnold ("Mathematical Methods of Classical Mechanics" is my textbook of choice for classical mechanics, available in English courtesy of Springer)
Walter Thirring (author of four-volume course on Mathematical Physics, English translation is widely available)
Walter Greiner (his series in theoretical physics is pretty much Lew Landau's Course in Theoretical Physics written in a light tone of Griffths textbooks, German original, all or almost all got translated to English)
Krzysztof Maurin (Polish original, translated to English. Very challenging three-volume Calculus course, standard in Polish physics curriculum until recently if I've had my sources correctly. To be honest, I would fear and respect equally anyone who went through it straight after high school)
Helena Rasiowa (Another Polish author, main book I know of hers is "Mathematics of Metamathematics". In my opinion THE book on modern set theory and abstract algebra)

If you would like something more, feel free to badger me here or via private messages. I'm happy to give some guidance or try to add something more your needs :D.

EDIT: Maybe to note for future reference, these are just my personal picks. They are not without their faults, some have rather old-fashioned notation (Maurin and Rasiowa, originals are from 1950's/1960's if I'm not mistaken) or tend to have serious examples/problems deficiency (Arnold, although Kotkin and Serbo problem set for classical mechanics, if possible to get in USA, is massively alleviating this problem). In addition, most of the books by the above authors are assumed to be on a graduate level by western publishers. That last point is largely a result of few last chapters being too advanced even on senior undergraduate honours courses… while having the rest of the book largely viable below this point. "Mathematics of Metamathematics" has a reputation (or so I was told by my Polish friend) of being a book that can be just as challenging to a high school student wanting to learn more and to a maths Ph.D who picks it to get some insights to a more advanced proof. It is wise to give a look to the bibliography and other references while reading and look for an errata in case of older editions.

On the other hand, same points apply to Feynman's Lectures in Physics, so it's not worth to stress over it ;).