A rigorous calculus self-study

[colonelcrayon]

Okay, here's the deal: I am going to run out of math classes at my high school at the end of this [junior] year. I'll have finished AP Calc BC (with Larson/Edwards as the text), but I already studied almost all of Gilbert Strang's free online text. So I'm really bored redoing material I already know. I want to do a rigorous self-study next year (maybe starting this year if I stay this bored). I want to work with a book that emphasizes theory and proof. I love the theoretical aspect and it would be great to get a taste of "real" math before heading off to college as a potential applied math major. My background is fairly standard USA K-12 math, with more rigorous intro calculus thanks to the Strang self-study.

My requirements for a book:

1) It must emphasize proof and have challenging material.
2) I would prefer if explanations are somewhat readable - über-formal prose can be frustrating.
3) Problem sets are a must, since I need to work through exercises to understand material properly.

The big three calculus/intro analysis texts (Spivak, Apostol, and Courant) are all obvious possibilities. Unfortunately, they are all really pricey. I could probably swing a copy of Spivak, but all those cheap Dover books have spoiled me.

What would you suggest?

 

[thrill3rnit3]

Since you're done with calculus, you should probably get started with an intro to analysis text.

I made a similar thread for a "cheap" text here:

https://www.physicsforums.com/showthread.php?t=333250

If you do have the chance though, get Spivak's.

 

[Petek]

I suggest that you look at https://www.amazon.com/dp/038790459X/?tag=pfamazon01-20 by Kenneth Ross. It's an inexpensive introduction to Real Analysis (= Calculus with proofs). It's good for self-study since it's almost free from errors and typos (having gone through many printings) and has solutions to some of the problems. Read the reviews on Amazon for more opinions (keeping in mind that some reviews tell you more about the reviewer than the book!).

 

[colonelcrayon]

^ Thanks for that recommendation.

I feel uncomfortable with my proof and problem solving skills. While I have the rote knowledge to push ahead, I worry that I might actually do myself a disservice. Maybe I should spend the next year focusing on problem solving and logic. I thoroughly enjoyed Polya's How to Solve It, but there were too few examples/exercises and I really need a proper textbook to study from.

One book that seems to combine the two goals nicely is https://www.amazon.com/dp/0131481010/?tag=pfamazon01-20. Unfortunately, it costs $85. Is there a cheaper book that might do the same?

 

[Petek]

I think it's good idea to improve your problem-solving skills. There are plenty of problem collections at almost every level. Also, next year you might consider taking a class at a JC/College/University near you. (I also ran out of math classes in high school and this is what I did.) A teacher or adviser should be able to help you to do so.

Finally, you might want to consider studying an area of math other than real analysis. A class in differential equations is usually the next step after calculus, and would be essential if you plan to major in applied math.

HTH

Petek

 

[colonelcrayon]

Also, next year you might consider taking a class at a JC/College/University near you. (I also ran out of math classes in high school and this is what I did.) A teacher or adviser should be able to help you to do so.

I'll look into it, but the logistics will probably be impossible. I am working through AMC/AIME, Euclid, and USAMTS problems, and I have a copy of the USSR Olympiad Problem Book (which is great).

I don't know that I would really benefit from a differential equations study at this point, since I'll end up taking it in college anyway and I would prefer to work on improving skills that will be useful in any math class (proof, logic, etc.)

 

[JG89]

I don't remember Courant's "Introduction to Calculus and Analysis Volume I" being too expensive. I think I got my copy off of amazon for $45 or so.

As for an introduction to proofs, I think the best thing you could do is to just read a crap-load of proofs until you get the hang of it.

 

[thrill3rnit3]

I'll look into it, but the logistics will probably be impossible. I am working through AMC/AIME, Euclid, and USAMTS problems, and I have a copy of the USSR Olympiad Problem Book (which is great).

How are you doing in those, if you don't mind me asking?

I don't know that I would really benefit from a differential equations study at this point, since I'll end up taking it in college anyway and I would prefer to work on improving skills that will be useful in any math class (proof, logic, etc.)

Definitely agree with this statement.

 

[Sankaku]

Instead of Analysis, try a good course on Linear Algebra. It would be good for someone at your level and a decent course will make you do tons of proofs. It would also be a change from the Calc you have been doing and show you some of the abstraction that is coming in the rest of Mathematics...

Try starting with:
http://tutorial.math.lamar.edu/Classes/LinAlg/LinAlg.aspx
and
http://joshua.smcvt.edu/linearalgebra/

 

[thrill3rnit3]

Agree with the recommendation of Jim Hefferon's notes, it combines the computational and the abstract part of LA pretty nicely for a first exposure.

That's where I first learned LA from, and I thought it was pretty good.

 

[colonelcrayon]

How are you doing in those, if you don't mind me asking?

Not too bad. I never got my AMC score from freshman year (?) and I was out of town last year, but I hope to do well this coming February. I did a Euclid with the timer last week and did fairly well (I don't remember the score off the top of my head). I fail at the USAMTS, but it's still fun.

As for the Soviet problem book... it depends. Many of the problems are truly insane, but I really like the proofs/solutions and it's fun to try a problem, fail, and when reading the solution get that "Aha!" moment when you see how they did it.

Instead of Analysis, try a good course on Linear Algebra. It would be good for someone at your level and a decent course will make you do tons of proofs. It would also be a change from the Calc you have been doing and show you some of the abstraction that is coming in the rest of Mathematics...

That's what I thought! I will definitely take a look at those links, but I learned during the Strang and Keisler self-studies that staring at a computer screen gets really old.

As far as an LA text is concerned, I know that Shilov's is supposed to be amazing once you have lots of experience (which I definitely don't). Is Axler a good choice for a beginner?

 

[thrill3rnit3]

As far as an LA text is concerned, I know that Shilov's is supposed to be amazing once you have lots of experience (which I definitely don't). Is Axler a good choice for a beginner?

I'd suggest learning from an elementary text first (Hefferon). That's how I learned LA, I started with Hefferon text's, then after that I moved on to Hoffman and Kunze's text, which is the abstract LA text.

 

[Reedeegi]

Though it may not be analysis, one topic that you may want to learn is axiomatic set theory. The discipline emphasizes proofs from axioms and trains you to think in a very abstract, proof-based way that would benefit you in all of your math studies. Luckily, Dover publishes, in my humble opinion, the best introduction to AST on the market in "Axiomatic Set Theory" by Patrick Suppes.

One way I learned mathematics was also to constantly challenge myself by giving myself the "impossible" texts instead of the "advanced" or the "beginners." If you wish to do Analysis, I'd recommend using Royden's "Real Analysis" as opposed to Spivak or Apostol. By using such a text you're forced to learn the fundamentals efficiently and with a goal as well as constantly challenge yourself.

I've been doing this since 8th grade, when I had the incredibly amazing opportunity to skip Algebra 1, 2, Geometry, and Precalculus and go straight into my local college's Calculus class, where the teacher emphasized proofs. I was forced in one week to lerarn essentially 4 years of material and (somehow) I succeeded by attaining a 4.0 in the class, where the average grade was ~2.5. The learning method is similar to that of language immersion, where you're forced to learn in order to fend for yourself.

Of course, this method does not work for everyone; but for the people who can pull it off, it works wonders.

 

[colonelcrayon]

^ Thanks for both of those. I will check out all books mentioned.

I have plenty of time on this, since I'm doing a chemistry self-study combined with contest prep this year. I won't be attempting a full load of self-study math until at least next spring.

 

[RedX]

Here are some online videos of math courses:

http://www.uccs.edu/~math/vidarchive.html

It requires free registration to view.

There is one course called "real analysis" but is really a course in "measure theory". If you want to watch this video however, the first two lectures of the class are missing, so you'll have to fill in some details as the videos start from the 3rd class onwards. It can be completely filled by learning what the supremum and infimum mean, and what the lim sup and lim inf of a sequence are. If you've had no exposure to limits of sequences and open sets and stuff, then there are other courses called "modern analysis" which I think cover those things (not sure though I've only watched the "real analysis" course).

 

[Sankaku]

Here are some online videos of math courses:

http://www.uccs.edu/~math/vidarchive.html

It requires free registration to view.

Wow! That is a huge archive of videos. Thanks for the link!

I am going to post a link in the Learning Materials section so other people can find it...

 

[colonelcrayon]

I will look at those. Thanks for the suggestion.