Vector calc vs. differential forms, a good textbook?

  • #1
Hello everybody,

This is my first time on Physics forums. I am a sophomore in high school who LOVES math. I have lots of free time this summer and would like to learn multivariable calculus and/or linear algebra (whichever is a prerequisite for the other, depending on the textbook I choose). I have two issues that each have a few questions:

I'm currently going through Spivak's Calculus at the moment - it isn't hard but it isn't easy either, which is a very happy medium for me (I feel appropriately challenged). I'm curious if there are good multivariable calculus textbooks that teach in a similar way (known for challenging material and are kind of chatty (not super terse)). Before you redirect me to a different thread, I have read them.

I've read that there are some books that are very good and teach a thing called vector calculus (which I've also read is the standard treatment of multivar). I've also read that there are other textbooks that teach a thing called differential forms which I've read are simpler, more elegant ways of dealing with the standard material, but are slightly unconventional.

Lastly, I WANT rigor but at the same time it would be great if I could get something more than just a theorem-book. I would like a book that discusses applications or at least physical intuition, if possible. Above all, I don't want some formula-book for people who don't care about proofs (nor do I want a book that lands in between theorem-book and engineer's guide to math). Is there some godly tome that combines both pure mathematics and interesting applications of theorems?

To end the rambling, here are my questions:
For someone who wants a mathematical challenge with some applications (if possible), what multivariable calculus text would you recommend? Should I study vector calc or differential forms, or both? If the third, which should I do first? Should I learn linear algebra beforehand, or does this depend on the text? If the text requires some knowledge of linear algebra, could you let me know?

I've attached a poll (although I'm not sure how it works or if it will), so that you can possibly nominate good textbooks and vote on them.

Thank you for your kindness and advice,

Answers and Replies

  • #2
You could read Spviak's "Calculus on Manifolds or Munkres' "Analysis on Manifolds"
Both books starts with a rigorous treatment of multivariable calculus and then devotes the latter half on introducing differential forms and manifolds.

You will need a very solid understanding of Calculus (At the level of Spivak should be fine)
and linear algebra before reading these books.
If you don't already know linear algebra, you would probably be better off spending your summer studying LA (say from Friedberg).
That would probably keep you busy for a few months.
  • #4
I would recommend going with linear algebra next, but that's just my opinion ... I have a little bias towards learning algebra after you get through something like Spivak, before you move on to multivariable calculus, vector calculus, and differential equations.

That being said, I would recommend buying an old copy of Elementary Linear Algebra by Anton. I used the 8th edition, which was kinda new when I started to learn LA, and it served me well. I'm sure you can find 7th or 8th edition on Amazon/Half/whatever for a few dollars since it's about a decade out of date. It's a good mix of theory with some application. There are enough "easy," algorithmic problems after each section to serve as a check that you're able to do computation and that you understood the material. There are also a few problems in each section that are more proof oriented. There are also problems that involve using calculus, which is great too since you're pretty much through Spivak.

Anton does author a book on applications which is generally for engineering math classes, so just make sure you're getting the one that's a bit more theory, else you'll probably be disappointed with the rigor.

I also highly recommend watching Prof. Strang's Linear Algebra lectures on MIT's OCW; they are fantastic. Watching those will serve as a wonderful supplement to your book learning and may help you look at algebra a slightly different way than what Anton presents it.

Prof. Strang authors a book as well, which I'm sure would complement the lectures perfectly, but if I were to be learning the subject from scratch, I'd rather have a slightly different perspective between the book and lecture ... plus, you can easily tailor how you move through the book to what the MIT lectures are covering. I'm pretty sure as long as you cover the basics first (aka chapter 1 and probably 2 of Anton) you can just start the lectures with no problem as far as prerequisite knowledge goes.

Good luck if you end up studying algebra this summer.
  • #5
Thanks everyone!

Micromass - if I went through this and did all of the exercises and whatnot, would this cover the same material in a multivariable calculus course plus an introductory course to linear algebra (plus some introduction to differential geometry)? I looked on amazon and I've already read rave reviews, but at the same time it was mentioned that this was used by a few people to teach high school students, implying it wasn't super in-depth.

The other question I have is, what's the deal with the author inventing his own notation in the book? A few reviews mentioned that. It's fine with me if he does that a little bit, but I don't want to be discussing math with someone and have to explain what I'm doing to them every time I use an unconventional symbol.

Otherwise, I'll look at the other textbooks but this looks good as it gives all the subjects in a unified manner. Again, thank you!
  • #6
Micromass, the other question I had was, does this book cover determinants and working with matrices, or is it more of just a pure math book?

For example, I read about Axler's Linear Algebra Done right, and it was explicit that they don't cover determinants until the end and try to avoid matrices. Not that mathematical elegance is a bad thing - I really want a good math book! I just want to learn the material that is applicable in the physical sciences along with that which is applicable in mathematics (if possible). It would be a shame if I couldn't use my linear algebra knowledge in a non-math context.
  • #7
Sorry I didn't include this in my first post, but Advanced Calculus of Several Variables by Edwards is a great book that has everything you're looking for. It'll be a great fit since you are coming from a background of Spivak for calculus but I still maintain that you should fit in some linear algebra (perfect for the summer before you move on to something like Edwards next school year perhaps?).

LA Done Right is an OK book for quite a few reasons, I have looked at it before but it just never seemed to "fit" with my progression. Anton or Strang are good first books that have solid theory but also have some application. They both segue nicely into Algebra by Artin, which is another one I'd recommend if you continue with algebra instead of more calculus before university ... then Roman's Advanced Linear Algebra segued nicely from Artin, for me at least. A few of my friends/acquaintances have given strong reviews of Axler's book but they've commented that it's almost for if you were exposed to an algorithmic engineering section of linear algebra that didn't involve much theory and then needed to get "caught up" later on ... which wasn't the case for me since Anton >>> Artin >>> Roman worked just fine for me.
  • #9
lastly, is hubbard as challenging as spivak?
  • #10
lastly, is hubbard as challenging as spivak?
Practically everything is proven in that book although that hardest proofs are in the appendix. So you could choose to treat this as a real analysis book, which is a step up from Spivak, if you choose to go over the proofs in the appendix, or not, and if you skip a few sections here and there, it would be about the same level as both Spivaks, but it would be much less dense than Calculus on Manifolds, even though is covers the same material and more.

Also, I really recommend you get the solutions manual since it is the best S.M that I have ever seen since it goes through all the steps of solving the odd numbered questions, even the questions that ask for a proof.
  • #11
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Homework Helper
i don't know if you care about money, but spivak is $45 for 150 pages, fleming is $57 for 400 pages, and hubbard is $99 for 800 pages.

I myself would go with fleming, but you may like to spend more and get more explanation. Or you may like a brief treatment.

Note that the books differ, since the two longer ones treat lebesgue integration, while spivak only does riemann integration.
  • #12
i don't know if you care about money, but spivak is $45 for 150 pages, fleming is $57 for 400 pages, and hubbard is $99 for 800 pages.

I myself would go with fleming, but you may like to spend more and get more explanation. Or you may like a brief treatment.

Note that the books differ, since the two longer ones treat lebesgue integration, while spivak only does riemann integration.
I'm going to go with hubbard, because I don't know linear algebra yet and I'm not mathematically mature enough to really enjoy rudin-style "theorem-proof-corollary" type dry material and would prefer a little wordiness.

Thanks for everyone's advice!
  • #13
Hey there,

I meant to ask one last thing: I read that the proofs in the book are done via algorithms. I've never seen that done before, nor do I really know what that means. (I know what an algorithm is and how one proves that it works, but how does one prove a statement via an algorithm?).

Did this at all trip anyone up who read the book, or is it an innovative and intuitive way of proving things?
  • #14
Proving something with an algorithm doesn't just prove that a solution exists(which is a common way to prove something), but also gives you a way to compute the answer as well. For example, the proof that ax^2 +bx +c = 0 has zero, one or two solutions is proved using an algorithm which gives you a way to find x instead of just showing that such an x exists.
The book uses almost no proofs that merely state the existence of a solution and instead uses proofs, some of which may be a bit more complicated, that allows you to find the solution. For example, in 2.10 of Hubbard, you see the Inverse Function Theorem which is proved in the appendix. It is more complicated than the proof found in Rudin; however, the proof in Hubbard also gives you a radius around a point in which you know that the inverse is invertable, which makes it more useful for computations.
  • #15
Score! That is the COOLEST THING!

This book seems to be exactly what I wanted - mathematical rigor coupled with applicability to the sciences.

Thanks so much!
  • #16
I liked "Advanced Calculus" by Buck. It is fairly in-depth, uses a visual/intuitive approach to motivate proofs, and doesn't require a whole lot of background. It also has plenty of applications.
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