How Should I Self-Study Math After Calculus II?

[MJC684]

I'd like to study math on my own at home so that it will make me a better student of mathematics. I am in college and have recently finished Calc II. I got an A in both calc 1 and 2 but yet I am unsatisfied with my knowledge and skill with the subject. My goal is to prepare myself for the advanced and proof based upper level courses. I want to be well prepared that is. Ahead of the game if you will.

I have access to plenty of undergrad math texts but I don't know where to begin. I want to spend my self study time wisely because I don't always have too much of it. For instance should I start with restudying calc 1 and 2? Would I use the same text or should I use a more rigorous one? But that leads me to my next question about proof writing. A more rigorous treatment of calculus will call for a lot of proof writing in the exercises and that is a skill I do not have. I never had plane geometry in high school actually but I do have an interesting intro proofs text that teaches basic geometry at a higher level. Its a gem called Introduction to Proofs: Basic Concepts of Geometry. Its sweet. So should I start out with learning logic,sets, and proofs first? I am already struggling to prove basic set laws like DeMorgan's and such. I need help! Where should I begin? Any and all sugestions would be appreciated.

 

[n1person]

One book that I particularly like is called "A Concrete Introduction to Higher Algebra" by Lindsay N. Childs. One of my friends used it for an introduction to proofs type class, and looking through it, it seems to be pretty reasonable and useful. It starts with constructing rational numbers, then induction, and then through lots of other useful topics, including introduction to analysis, linear algebra, and group theory.

 

[MJC684]

Oh yea I've seen this book in the stacks at school. I could definitely use some help with induction. I'll have to check it out. Thanks man.

 

[micromass]

Try a proof book like the one by Velleman.

To be honest, proving things is quite a skill, and you can only learn it by doing. You'll need to read a lot of proofs and you'll need to try those proofs yourself.

You already know calc, that's good. Now, try to find a rigorous treatment of calculus (one that involves proofs) and study that. I recommend Spivak's calculus. Work through it and solve all the exercises (which tend to be difficult), you'll know proofs in no time!
Another option is to get an easy real analysis book and study from that. "Understanding Analysis" from Abbott is a very good book and it will get you acquainted with proofs. And knowing some real analysis certainly gets you ahead of the game!

 

[romsofia]

I would suggest learning vector calculus, multivarible calculus and ODE's then diving into a proof based calculus book.

 

[QuarkCharmer]

Not that I, in any way, relate to you, though we are around the same math level (course wise at least). I have little knowledge of proof based mathematics, and worked through the first two chapters (not even the calculus part!) of Spivaks book doing all of the problems and it was a monumental task. I sincerely doubt this is the best way to go about learning proof based mathematics.

I have a copy of the PF recommended "How To Prove It (Velleman)", and it's pretty difficult for me as well. I signed up for Linear Algebra next semester, and I am hoping that course will give me the ability to at least get further than a few chapters in Spivaks and the Prove It book.

Is there perhaps another, easier book?

 

[micromass]

Not that I, in any way, relate to you, though we are around the same math level (course wise at least). I have little knowledge of proof based mathematics, and worked through the first two chapters (not even the calculus part!) of Spivaks book doing all of the problems and it was a monumental task. I sincerely doubt this is the best way to go about learning proof based mathematics.

OK, Spivak's exercises are very difficult. But I like the theory in Spivak.

I have a copy of the PF recommended "How To Prove It (Velleman)", and it's pretty difficult for me as well. I signed up for Linear Algebra next semester, and I am hoping that course will give me the ability to at least get further than a few chapters in Spivaks and the Prove It book.

Is there perhaps another, easier book?

Proofs must be done in a context, just reading a proof book won't teach you proofs. It's only handy if you already know how to do proofs, but if you just want to expand your knowledge.

Try "Understanding Analysis" by Abbott. It's a very accessible book with exercises that are not too difficult. The book is written for people that know calculus, but who aren't used to formal proofs. So the book is actually written for people like you! Try it out...

Since you're going to see linear algebra soon, why not pick up a linear algebra (proof-based) book and study from that now??

 

[MJC684]

I actually have the Spivak calculus text at home collecting dust on my shelf because at my skill level I felt that the enormous amount of time at it would take me to work through a single chapter of spivak would be better spent on something not quite as rigorous but I could completely wrong. I also have Apostol Calculus Vol 1 which looks awesome and can't wait to sink my teeth into when I'm ready.

Actually Micromass I was thinking about starting my quest with Linear Algebra. Does anyone feel that it is smarter to take Linear Algebra before Multivariable/Vector Calc? Should I start Linear Algebra with a proof based book? Or something milder like Larson's or Anton's Elementary Linear Algebra? I also have this also book called Calculus Two: Linear and Non-Linear Functions which integrates Linear Algebra into multivariable/vector calculus. It also calls for proofs in the exercises. Is that a smarter move? I know its not a replacement for dedicated Linear Algebra course which I will need regardless. My ultimate goal is to be able to study Differential Geometry/Manifold Theory. I vector calculus by Marsden and Tromba which seems to be more advanced than other vector calculus texts.

Also, I have How To Prove It and many many other Intro Proof/Logic/Sets texts. Any other opinions?

 

[Herricane]

I also have Spivak's calculus collecting dust in my bookshelf. I wish there were other people that wanted to work through it too. I like working individually, but it is nice to have other people working with you too.

 

[micromass]

I actually have the Spivak calculus text at home collecting dust on my shelf because at my skill level I felt that the enormous amount of time at it would take me to work through a single chapter of spivak would be better spent on something not quite as rigorous but I could completely wrong. I also have Apostol Calculus Vol 1 which looks awesome and can't wait to sink my teeth into when I'm ready.

Actually Micromass I was thinking about starting my quest with Linear Algebra. Does anyone feel that it is smarter to take Linear Algebra before Multivariable/Vector Calc? Should I start Linear Algebra with a proof based book? Or something milder like Larson's or Anton's Elementary Linear Algebra? I also have this also book called Calculus Two: Linear and Non-Linear Functions which integrates Linear Algebra into multivariable/vector calculus. It also calls for proofs in the exercises. Is that a smarter move? I know its not a replacement for dedicated Linear Algebra course which I will need regardless. My ultimate goal is to be able to study Differential Geometry/Manifold Theory. I vector calculus by Marsden and Tromba which seems to be more advanced than other vector calculus texts.

Also, I have How To Prove It and many many other Intro Proof/Logic/Sets texts. Any other opinions?

Doing linear algebra before multivariable calc is a smart move. You'll need some linear algebra to study multivariable calculus.

The faster you start with proofs the better and the more proofs you do the better. So I would highly recommend to get a proof-based book on linear algebra, and not something milder. Linear algebra is the perfect topic to be confronted with proofs!

 

[MJC684]

Alright sounds good. Linear Algebra it is then.

 

[BrianMath]

The Linear Algebra text that I'm using to teach myself is http://joshua.smcvt.edu/linearalgebra/" , available online for free.
One thing I thoroughly enjoy about this text is that it has solutions to every problem, which helps when "grading" your own work.
This also means that if you see problems that look interesting but just out of your reach, you can get a head start into the problem in the answers manual and go from there.

It is supposed to cover a full first semester of Linear Algebra, from linear systems and Gaussian elimination to Jordan Canonical Forms. I'm just about finished (currently on the final section of the 5th chapter), and I love the way everything follows from previous topics, like how performing row or column operations on a matrix is mechanically the same as multiplying by matrices.

Something like permuting two rows
$$\begin{pmatrix} a & b & c\\ d & e & f\\ g & h & i \end{pmatrix}\xrightarrow{P_{1,2}}\begin{pmatrix} d & e & f\\ a & b & c\\ g & h & i \end{pmatrix}$$

is the same as left multiplying by a permuted identity matrix

$$\begin{pmatrix} 0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1 \end{pmatrix}\begin{pmatrix} a & b & c\\ d & e & f\\ g & h & i \end{pmatrix}=\begin{pmatrix} d & e & f\\ a & b & c\\ g & h & i \end{pmatrix}$$

and similarly with column operations by right multiplying. It is all so intriguing and wonderful, and I've learned much more from this than I did in the course I took last year.

There are a few errors, but not many, and I've sent a few emails to the author and he's added them to this summer's revision list once he gets a chance to get back to it and correct them. They aren't really bad mistakes, either, the worst I would say would be an incorrect sign on one of the answers in the Cramer's Rule section.

 

[QuarkCharmer]

Nice, I like how he has a recommended schedule for the sections of the book, and that it is split into sections for those who have only had a semester of calculus, and those who have had multi-variable.

I'm going to start looking through this one next week!

If I like this book, what are the options to get it bound? I can't stand reading from a computer screen for a long duration of time. Can I get something like that (450-500) pages bound in some form at like, kinkos?

 

[Bourbaki1123]

When I was a freshman, I snagged a ton of ebooks and cycled through them trying to do the exercises. I tried to attack Rudin's book after getting through calc 1 and then when that didn't work I tried again after calc 2, and I thought I understood more than I did (in my defense, I didn't take any math after sophomore year in high school, so I was perhaps unusually overzealous about 'catching up'). It turned out that most of the solutions I had come up with went wrong somewhere because I hadn't learned how to really write a proof, I just tried to jump in head first. What actually helped me out quite a bit was working through W.V.O. Quine's book "Methods of Logic" (I found that to be a very gentle and readable intro to first order logic) and Dummit and Foote's Abstract Algebra, which I found to be quite an easy read for a math text. Getting through both of those really brought me up to speed with the basics of proofs in a few month's time.

 

[snipez90]

Sh*t, there are like a thousand of these threads with more or less the same advice and sometimes just plain weird advice.

I'll emphasize once again that honing basic problem solving skills at your level is probably far more important than building basic proof writing strategies. In undergrad math courses, writing up a solution is something that you gradually learn to deal with, but it's far more troublesome if you have absolutely no idea how to begin attacking a particular problem at the end of a certain chapter.

Sure you can try tackling Spivak and all of its problems, but I guarantee you that if you don't have the problem solving ability that suffices for say, any entry level precalculus math competition (the AMC 12 is the notable one in the US), you will struggle with Spivak. You might understand everything in the actual text, since Spivak is an exceptionally clear and gentle expositor, but it's unlikely you'll be able to do a significant portion of the problems.

 

[HeLiXe]

Hi MJC684 :)

I found that the Thinkwell Calculus lectures by professor Burger make problem solving intuitive and the lectures are easy to comprehend. I have taken online courses using his lectures for Calc I and II.

I also want to do a more thorough review of proofs and the subject matter in general -- I still do not feel like I have mastered everything. Good luck and be sure to let us know what works for you!

 

[206PiruBlood]

Not that I, in any way, relate to you, though we are around the same math level (course wise at least). I have little knowledge of proof based mathematics, and worked through the first two chapters (not even the calculus part!) of Spivaks book doing all of the problems and it was a monumental task. I sincerely doubt this is the best way to go about learning proof based mathematics.

I have a copy of the PF recommended "How To Prove It (Velleman)", and it's pretty difficult for me as well. I signed up for Linear Algebra next semester, and I am hoping that course will give me the ability to at least get further than a few chapters in Spivaks and the Prove It book.

Is there perhaps another, easier book?

My experience has been remarkably similar! After reading recommendations on PF and Mathwonk's evaluation on Amazon I chose Spivak's Calculus as my first exposure to rigorous mathematics. Right away the problems were too difficult, and so at the recommendation of PF, I set Spivak aside for Velleman. Some chapters were more difficult than others but I managed to do about 80% of the problems. Frankly I find it so much easier to devote more time and attempt all problems during self study.

With a great deal of confidence I am now working through Spivak; in fact, many of the early problems from Spivak I had already completed while working through Velleman! The problems are still very difficult but they are now accessible and my appreciation for math grows every day.

As for the getting stuck in Velleman's book or any other really, how stuck do you get? Can you solve a significant number of problems from each section? I found if I just kept moving forward, skipping what I had to, I could return later and the problems weren't as daunting.

 

[MJC684]

Because people on this foum are always recommending Spivak to beginners and claim they worked through it before they had finished the standard calc sequence I always felt like maybe I wasn't cut out for Mathematics due to the fact I was no where near ready to attack it meaningfully. Am I the only one that feels that way?

 

[206PiruBlood]

Because people on this foum are always recommending Spivak to beginners and claim they worked through it before they had finished the standard calc sequence I always felt like maybe I wasn't cut out for Mathematics due to the fact I was no where near ready to attack it meaningfully. Am I the only one that feels that way?

I wouldn't let this experience hold you back. Consider that the book was written for an audience with a different background than is typical of today's high school graduates.

Do you have any experience with proofs? If not Spivak will certainly pose a monumental challenge. I had taken the entire calc sequence, differential equations and linear algebra before starting Spivak; other than a couple proofs, pretty much everything I did in those courses was unnecessary for tackling the problems from Spivak. My point is that this text requires a specific background, which you might not have - yet! Spivak is a geat introduction to rigorous mathematics, but it is necessary to get acquainted with the language first.

Why don't you try a book about logic, set theory, and writing proofs.

 

[MJC684]

Yea thanks bro I'll definitely check those lectures out.