Starting my Self-Study of Rigorous Mathematics

[Tee2612]

Hello everyone!

I'm new here.

I'm starting a self-study of more rigorous mathematics. My background is I have a B.S. in Mathematics(class of '14) and have had rigorous classes but they were, in my opinion, sub-par and not taught as rigorously as they should have been. Truth be told, I probably wouldn't have done that great if they were lol.

With that being said, I do have good exposure to calculus as far as computation(integration, differential, Differential Equations). Had a class on "proofs" but never really "got it" as deeply as I'd hoped.

With that being said I am self studying and am starting with these two books:

https://www.amazon.com/Concise-Introd...9443632&sr=1-2&tag=pfamazon01-20

https://www.amazon.com/dp/013897067X/?tag=pfamazon01-20

Does anyone have any advice on the sequence of courses I should study after these two books?

Should I study Spivak Calculus after Martin Liebeck's book on proofs or go straight to Gaskill's Real analysis?

 

[jedishrfu]

Your first web link to Concise is broken...

What is the goal of your studies? to remember what you were taught? to go on to grad school? for a job? for fun?

If you're starting with Calculus then would the next logical thing be Differential Equations, Linear Algebra or for a math major
Set Theory, Abstract Algebra, Point Set topology...

Here's an example of course sequence depending on your interest:

http://www.math.harvard.edu/pamphlets/courses.html

For cool proofs, there's Proof from the Book:

https://www.amazon.com/dp/3642008550/?tag=pfamazon01-20

 

[homeomorphic]

Personally, I'd do Visual Complex Analysis before those books. I attribute my earlier successes in proof-based classes to the semi-rigorous intuitive style of that book, plus my engineering background, particularly studying electromagnetism and signal processing. If you don't have good visual insight, it's hard to do real analysis proofs and complex analysis is a much richer playground for that than real analysis is. I am a bit of a fanatic when it comes to that book, and sometimes, I feel silly recommending as much as I do, but I think it's particularly relevant to "getting it", as you put it.

 

[Tee2612]

Your first web link to Concise is broken...

What is the goal of your studies? to remember what you were taught? to go on to grad school? for a job? for fun?

If you're starting with Calculus then would the next logical thing be Differential Equations, Linear Algebra or for a math major
Set Theory, Abstract Algebra, Point Set topology...

Here's an example of course sequence depending on your interest:

http://www.math.harvard.edu/pamphlets/courses.html

For cool proofs, there's Proof from the Book:

https://www.amazon.com/dp/3642008550/?tag=pfamazon01-20

https://www.amazon.com/dp/1439835985/?tag=pfamazon01-20

To help me understand proof better.

 

[verty]

This was posted in the other thread, I've moved it here.

Hello everyone!

I'm new here.

I'm starting a self-study of more rigorous mathematics. My background is I have a B.S. in Mathematics(class of '14) and have had rigorous classes but they were, in my opinion, sub-par and not taught as rigorously as they should have been. Truth be told, I probably wouldn't have done that great if they were lol.

With that being said, I do have good exposure to calculus as far as computation(integration, differential, Differential Equations). Had a class on "proofs" but never really "got it" as deeply as I'd hoped.

With that being said I am self studying and am starting with these two books:

https://www.amazon.com/dp/1439835985/?tag=pfamazon01-20

https://www.amazon.com/dp/013897067X/?tag=pfamazon01-20

Does anyone have any advice on the sequence of courses I should study after these two books?

Should I study Spivak Calculus after Martin Liebeck's book on proofs or go straight to Gaskill's Real analysis?

Hello and welcome. I think you must have enough knowledge not to need the first book, it seems to be a cookbook of random topics, probably not going too deeply into anyone topic. Gaskill also looks somewhat shallow, it has nothing on R^n which is unfortunate.

You did a proof course before, what you need to decide is at what level you are. If you are at a sufficient level to prove simple things, Spivak is probably the book you want. It is difficult, make no mistake, but it does present one with the need to prove things and that makes it good. In my opinion, it is great for that stage where you have the knowledge but aren't confident. But if you need a refresher, the Liebeck book looks fine.

You asked whether you should follow Liebeck with Spivak. I think it would help to do that. A shallow real analysis book is probably less good, I would think. But I don't regard Spivak as an analysis book, more a "how to do analysis" book. Any learning is becoming familiar with a body of knowledge, and from this point of view, Spivak is about learning to prove things. Once you have that working knowledge, it is a book more difficult than it needs to be, IMHO. So, if you are confident or if Liebeck is particularly good and you feel ready to skip the training phase, you can skip Spivak.

Would Gaskill work as an alternative to Spivak? Yes, I think so, if you do the proofs yourself where possible. If you just read it like a book, that probably won't be sufficient preparation.

The sign that you are ready to skip ahead is if you are at the point where you can read theorems and proofs and picture in your mind what they are saying or what the relevance is. In other words, you can read theorems and proofs like a language and focus on the content of what is being asserted. Then you can say you are mathematically mature and ready to forge ahead.

I like this book for having a nice layout and selection of topics:

https://www.amazon.com/dp/0387974377/?tag=pfamazon01-20

That said, I'll quote one of the reviews to give you an idea of the type of book it is:

Please note this book is an analysis book not a calculus one, this means you will find a lot of proofs, theorems and definitions but not many exercises to learn to make calculations.

 

[Tee2612]

Gaskill's book came in the mail yesterday and I've read chapter 1 of Liebeck's books, its pretty good. I'm trying to get a feel for what proof techniques to use on what types of problems, although more than one can be used. And I'm trying to follow the logic. His book is really good and it spoon-feeds you as well.

The reason why I chose Gaskill is because one of the reviews said it's a "spoon-feeding" introduction to analysis which is what I think I need before I get into harder treatments of abstract mathematics.

 

[homeomorphic]

My approach to real analysis was not so much proof-techniques. It was mainly visualization and drawing pictures, and then translating that into proofs. My grade was over 100% the first semester with extra credit, so I must have been doing something right. Wasn't the hardest class it could have possibly been, but everyone else seemed to be struggling. You are right that you need to work on proof techniques/logic to some extent because you need that for the translation step and sometimes in coming up with the argument, but that's of secondary importance, unless it is exactly where the gap in your skills is, which could be the case, but is probably not the main difficulty most people have.

 

[Tee2612]

I'm going to brush up and refresh my algebra and Trigonometry skills with

https://www.amazon.com/dp/0878915087/?tag=pfamazon01-20

https://www.amazon.com/dp/0878915567/?tag=pfamazon01-20

Then I will move on to Spivak.

I also have one for calculus, advanced calculus, differential equations, linear algebra, and Probability.

 

[Tee2612]

Should I master Linear Algebra before I restudy calculus from Spivak?

I have Three Linear Algebra books I ordered:

https://www.amazon.com/dp/0763782491/?tag=pfamazon01-20

https://www.amazon.com/dp/0130470171/?tag=pfamazon01-20

https://www.amazon.com/dp/071674337X/?tag=pfamazon01-20