Calculus and Linear Algebra for Self Study

[oakleykid]

Hi all!

I am a current senior in high school, and will be finishing BC by the end of this month. I do not yet know where I am going to college, but I am planning on trying to place out of a few math classes before arriving. II will most likely major in engineering, but I still am interested in rigorous math (math minor, perhaps?)

Obviously BC is by no means rigorous, so I was thinking of studying both volumes of Apostol for single and multivariable calculus and linear algbera. I have heard Apostol is a bit boring, and Spivak is more interesting, but then I would need a different book for multivariable. I also have Shilov's Linear Algebra book, which seems pretty good. Hopefully I will be able to start with differential equations or analysis in freshman year.

So, are there any other books that would work for my purposes, or I would I be best using Apostol and Shilov?

Thanks!

 

[Fredrik]

I think any of the standard texts will be fine. If you want more detailed comments, I suggest that you use the search feature first. You're not the first person to ask about books on calculus and/or linear algebra. If the threads you find don't answer all of your more specific questions, then ask them here.

 

[oakleykid]

Thanks !

Just one question... Is Apostol's treatment of linear algebra sufficient or would it be best to use another book (Shilov, or Linear Algebra Done Wrong looks good) instead?

 

[osnarf]

Hi there, welcome to PF.

I'm in the same boat, I'm studying Engineering and Physics, and am learning to like more rigorous treatments of mathematics as opposed to just knowing how to apply it.

First off, have you had any experience with proofing? If you have, skip this paragraph. In high school i know i didn't. I still didn't when I started Spivak, so even after taking the calc series in college it was hard initially to self study a proof based book. Point being, your time would probably much better spent picking up a logic and proof book than jumping right into spivak or apostol. Like I said, I didn't, and I made it through the book, but initially it was very tough and I wound up getting a solutions manual and glancing at it one line at a time when I got stuck, because all my proofs were flawed. About 1/3 of the way through I didn't need the solutios manual any more, but I would have gotten much more out of the first 1/3 if I didn't need to use it.

I highly recommend Spivak, I read the first few chapters of Apostol V1 and Spivak's Calculus and much preferred Spivak's style. I suggest you do the same, just go to the library and read through a little of both and see which you like better. I don't think Apostol's book seemed that 'dry' (see https://www.physicsforums.com/showpost.php?p=1204541&postcount=14), but it is a little less conversational, if you know what I mean. I particularly liked how spivak leaves so much of the subject for you to figure out with carefully guided problems. As such, it's important you do EVERY problem. You won't find them boring, they're not the usual evaluate this integral or differentiate this function, with the exception of the first problem or two in a few chapters. As an aside, i particularly liked Spivak's chapter on planetary motion.

Again though, it's just a matter of preference, they contain roughly the same information.

Don't fret about needing another book for multivariable calc, I checked out Spivak's Calculus on Manifolds 2 weeks ago. The book is ~ 130 pages long, so I am just finishing up chapter 2 / 5. So far I have found it to be excellent, and I have a much better understanding than I did after Calc 3. It has the additional benefit that I am 2/5 through and it has only been 2 weeks (although I've dedicated a lot of time to it). I doubt the same could be said Apostol V2, Calc on Manifolds is very dense with information, you will need a firm grasp of at least the basics of linear algebra for this book, though. Again, do ALL the problems.

As for the linear algebra, if you are able to follow along with Shilov then I strongly recommend that - I'm also currently reading this. It goes much deeper than the $100 book they had me buy for my class. If it seems too much for now, I hear good things about Anton's book. Again, though, if you are unfamiliar with math logic I would grab a book on that and work through it before you get into this, it will make your life a little easier. Not as necessary as for Spivak, though. He really takes it from the top of linear algebra and works it through all the important proofs with you, as opposed to leaving them for the exercises (with a few exceptions). Maybe familiarize yourself with the basics of matrices first, multiplication, addition, etc.

Good luck

 

[oakleykid]

Thanks for all of the info!
My experience with proofs is pretty limited. Just HS geometry (I took it outside of school, and the teacher was a math phd, so it was better than average), and what I know from a theory of computation book I am reading. I think I could definitely benefit from logic/proof book.

I already have a basic background in Linear Algebra: matrix operations, eigenvalues, and a little bit of the SVD theorem, so I think I will stick with Shilov. I also have Anton's Calculus book, which has a few chapters on vector calculus. I assume this could be useful to reference if I get stuck on some of the basic concepts of multivaribale.

After reading from both Spivak and Apostol, I think I'll go with Spivak. Reading him will probably make studying all of this on my own a bit more enjoyable!

So, is this my basic sequence?
Intro Logic book --> Spivak Calculus --> Shilov Linear Algebra (or at the same time as Calculus?) --> Calculus on Manifolds

For the logic/proof book, should I be looking for an introduction to logic type book, a "how to prove it" type, or something altogether different?

Also, just for reference, how long did it take you to get through Calculus? I will be starting in March, and I hope to finish all of these over the summer (and I don't have much else to do!).

Thanks again!

 

[Fredrik]

You don't need a whole book on logic. You just need to study things like truth tables for logical operations, so that you understand e.g. that when a book asks you to prove that "A implies B", you can prove "not B implies not A" instead. I assume that this is the sort of thing that "how to prove it" books cover, but I have never actually looked inside one.

A difficult book on calculus, one that covers difficult integration problems, how to find limits of sequences, convergence criteria for series and generalized integrals, etc., will probably take you at least two months to get through, probably more, since you're studying on your own.

You can study linear algebra and calculus at the same time.

 

[oakleykid]

I think I'll go with some sort of how to prove it type book, so Amazon is probably my best bet.

I assumed Calculus would take a few months, so that's fine with me. Hopefully, doing it alongside Shilov's Linear Algebra will give me even more experience in proofs, and more preparation for Calculus on Manifolds, if I get there before the end of the summer. Thanks again.

 

[osnarf]

Yeah, that book would work fine, I liked the chapter on induction. Spivak took me about 2-3 months, I think. There's lots of problems!

You can definitely do Shilov and Spivak at the same time; neither one really needs the other (as far as I've gotten in Shilov). Shilov assumes some basic calculus in a few parts, but nothing you won't know from AP calc.

 

[oakleykid]

Ok, sounds like I have a plan!

Thanks for all the help guys!