Hard time starting analysis, want to start fast

[Exalt]

Back in the end of october, I picked up math as a hobby (although it will also help me career-wise, but in the end I probably won't go into a math related career, unfortunately. Currently my major has nothing to do with math). I had only completed up to calc II for the AP test, and hadn't done proofs since geometry 5 years ago. I started by picking up courant's analysis. Though the reading was fine, and I understood the proofs (albeit very slowly), I had no idea how to do any of the exercises.

So I started off lower by going to Solow's proof book, and another called 100% mathematical proof. I went through them fine, though there were a few exercises I didn't understand, but the proofs were easy. Then I went back to courant. I still didn't understand it. The first section baffled me, and the solutions were totally bizarre. I went back to solow's book, with a section on 'dissecting proofs', and courant was still beyond me. And this was supposed to be one of the easier analysis books...

I'm kinda hysterical right now, because I spent 3 months feeling like I've wasted my time, which is running out. I'm not a fast learner so I need all the time i can get, but I'm very very very busy all the time.

What step am I missing? More proofs? Also, how long should each proof take me? I want to be able to finish chapter 1 before February, and the whole book before June. I want to try to get as much of a boost in math before September, because then I can only do math for 15 hours per week (the summer allows for 30 hrs/week). My schedule and finance only allows for a math minor (due to things i won't be discussing), and i want to try to have enough knowledge in 4-6 years as a math major so that I will have enough ability to get into a top #15-25 grad school. btw, this is for applied math, so I'm also trying to work on my programming, which is taking up even more of my time.

I'm going to be taking calc II again this semester, along with discrete math.

I heard mathwonk started calculus off with courant, so I'm interested in how that worked out.

EDIT: Am starting to look at harvey mudd's videos. i also want to keep at courant, since i don't have the resources to purchase another textbook.

 

[╔(σ_σ)╝]

Please clarify what you mean by...

...I had no idea how to do any of the exercises.

This could mean many things.

Do you mean you tried the excersices and had no idea what to do ?

Or do you understand what you need to do and have a hard time doing it?

If the first question is the case then perhaps you are reading too fast and not taking time to absorb the material.


So can you explain.

 

[Exalt]

Please clarify what you mean by...


This could mean many things.

Do you mean you tried the excersices and had no idea what to do ?

Or do you understand what you need to do and have a hard time doing it?

If the first question is the case then perhaps you are reading too fast and not taking time to absorb the material.


So can you explain.

i don't have my books with me right now, but (if anyone has courant) I tried doing the problems, but my answer was way off from the answer in the book, which appears to have some thinking that was taught from a lower level standpoint. the trouble is around 1.1 c problems.

the thing that confuses me is that courant is supposed to be elementary... so what could be more elementary than courant for analysis?

 

[╔(σ_σ)╝]

I don't own a copy of the book.

The issue is that I am not sure if you don't understand the concepts or if you don't know how to write proofs.

Which is it ?

Perhaps, you should consider a newer text.

 

[micromass]

I looked at the Courant book, and it doesn't seem very easy for the beginner. Are you trying to learn calculus or real analysis? If you're trying to learn calculus, then go for spivak's book, I think it's a better book. If you're doing analysis, then maybe you need some more experience with calculus first...

Anyway, one of the downsides of self-study is that nobody will explain anything if you don't understand. And nobody willl check your exercises for mistakes and stuff. That's why I highly recommend to make use of our homework forum. Try to solve the problems on your own, and if you meet difficulties: make a post on the forum!

Reading proof books can only do so much. The only thing you can accomplish with it, is to give you a general idea of what a proof is supposed to be like. Once you got this, you need to start with the course you want to study. Try to read everyhting, and try to understand everything. Math is a slow process, so this will probably take a lot of time, but eventually you will get there. And you'll get there even faster if you ask questions to the right persons!

So,try doing some more calculus first. And maybe read from an easier, more modern book. Once you've got the hang of it, you'll find it easier than expected...

 

[Exalt]

I don't own a copy of the book.

The issue is that I am not sure if you don't understand the concepts or if you don't know how to write proofs.

Which is it ?

Perhaps, you should consider a newer text.

more about the proofs

Also, is there a difference between calculus and analysis? i thought spivak and courant taught the same stuff.

if i go for a more modern book, should i read that and then a more advanced one, or is one book just fine?

 

[jgm340]

OK, I'll give you a few questions, and let's see if you can answer it. If you can't answer it without looking at a book, I'll just tell you the answer, and ask you a similar question.

Question 1: Is it possible for |x| to be less than 1/n for any natural number n, but for x to not equal zero?

Question 2: For any natural number n, is there a real number x such than |x| < 1/n?

Question 3: What is the rigorous definition of a function?

Question 4: What is the definition of an onto (surjective) function?

Question 5: What is the definition of a one-one (injective) function?

Question 6: What is the definition of a bijection?

Question 7: Is there a way to count all the rational numbers? In other words, can we assign a rational number to each natural number, such that every natural number is accounted for?

Question 8: Can we do the same for the real numbers?

Question 9: How do we know when an infinite sequence x1, x2, x3, ... converges to some number x? i.e., what is the definition of convergence of a sequence?

Depending on how you can answer these, I can give you proofs to try.

 

[micromass]

more about the proofs

Also, is there a difference between calculus and analysis? i thought spivak and courant taught the same stuff.

if i go for a more modern book, should i read that and then a more advanced one, or is one book just fine?

Ah, well there is a huge difference between calculus and analysis. Real analysis is really an upper level course, only taken if you understand calculus very well.
I really suggest taking a more modern book. I looked at Courant, and it really shows that it's an old book. And an old book mostly means: no conceptual explanation and hard problems.
Take Spivak's book instead. The first few chapters are quite understandable and it will give a gentle introduction to proofs. Not all the problems are easy, but there are enough problems which gives you time to adjust.

Once you've completed an introductory book, like Spivak, you could look for more advanced text. (and you should!) But first things first, start with a nice, conceptual book...

 

[ttyu6]

Question 1: Is it possible for |x| than 1/n for any natural number n, but for x to not equal zero?

Do you mean |x| less than 1/n?

 

[jgm340]

Do you mean |x| less than 1/n?

Haha, of course! Fixed.

 

[ttyu6]

Ah, well there is a huge difference between calculus and analysis. Real analysis is really an upper level course, only taken if you understand calculus very well.
I really suggest taking a more modern book. I looked at Courant, and it really shows that it's an old book. And an old book mostly means: no conceptual explanation and hard problems.
Take Spivak's book instead. The first few chapters are quite understandable and it will give a gentle introduction to proofs. Not all the problems are easy, but there are enough problems which gives you time to adjust.

Once you've completed an introductory book, like Spivak, you could look for more advanced text. (and you should!) But first things first, start with a nice, conceptual book...

Agreed. I tried self teaching Rudin and found it extremely difficult having had a weak background in proofs. After completing Spivak's calculus(we used Stewart for calc I-III), I found Rudin much more manageable.

 

[╔(σ_σ)╝]

more about the proofs

Also, is there a difference between calculus and analysis? i thought spivak and courant taught the same stuff.

if i go for a more modern book, should i read that and then a more advanced one, or is one book just fine?

Analysis is calculus with all the proofs and details. "Calculus" is computation like taken derievatives and finding limits. Analysis is more rigorous and answerw questions like what is a limit? , when is a function integrable? etc... questions you don't usually answer in "calculus".

A book like spivak is an analysis book in disguise! In fact, I have seen analysis books on the same level or slightly lower level than spivak.

The idea of a newer text is to cater to the needs of students of THIS day and age. A typical grade 12 student 20 years ago would probably posses more mathematical maturity than an average grade 12 student today.

That been said, you need to find a book that does not assume you have a high level of mathematical maturity to begin with.

A lot of newer text are written with the "todays" student ability in mind.

 

[micromass]

EDIT: Am starting to look at harvey mudd's videos. i also want to keep at courant, since i don't have the resources to purchase another textbook.

Resources are not really a problem, you can always check out a library. Or, if you want to be more devious, you can download a lot of books from filestube or rapidshare...

Also, check out Khan academy videos. They're great! But they won't help you prove things, though...

 

[╔(σ_σ)╝]

I myself went from stewart to elementary classical analysis ( Marsden and Hoffman). I am still reading it right now and it is a very good book. There are lots of examples and questions to build up confidence with. There are also lots of challenging problems.

I personally think that is one of the better analysis books I have seen.
I used Introductory Real Analysis by Frank Dangello for my calculus/intro to analysis course. The book is extremelyclear and the level is not that high; maybe around spivak, if not lower (in terms of number of challenging problems). Perhaps its clarity makes it seem not to advanced.

My only problem with the book is its outrageous price tag.