Progression to prepare for analysis

[system1834]

I am currently an adult student about to start a mathematics program at a university. I want to start preparing myself for analysis as soon as possible because I have a feeling it is going to be very difficult. So, I recently purchased Apostol's calculus book as well as Spivak's since I heard these are good books to get someone used to doing proofs. However, they seem to be a bit over my head. I am able to understand many of the concepts they are explaining, but when I get to the problems at the end of a unit, I don't have any idea where to start. Are there any books that would help get me to the level where I would be able to complete these two books?

 

[chiro]

I am currently an adult student about to start a mathematics program at a university. I want to start preparing myself for analysis as soon as possible because I have a feeling it is going to be very difficult. So, I recently purchased Apostol's calculus book as well as Spivak's since I heard these are good books to get someone used to doing proofs. However, they seem to be a bit over my head. I am able to understand many of the concepts they are explaining, but when I get to the problems at the end of a unit, I don't have any idea where to start. Are there any books that would help get me to the level where I would be able to complete these two books?

Hey system1834 and welcome to the forums.

I don't have much familiarity with those books but I am on track to graduate this year with a duoble major in mathematics so I will give you some advice on what I have learned.

The first thing, and the most important point for your question is that you build up intuition with mathematics.

For me a lot of actually doing this was with 3D programming. A lot of the geometric concepts were presented to me in a variety of forms and I had to learn a lot of techniques to build appropriate geometric figures and use them in various ways.

This kind of experience ended up making a lot of the geometric concepts as well making visualization of mathematics in general (not just things classified as 'geometry') a lot more useful.

This actually took me quite a long time to do and it was not just a completely directed exercise (we're talking more than five years here) so I can't exactly recommend doing what I did.

But what I can recommend is that you find whatever material you need to get an intuitive grasp before going to the proofs perspective.

Also with mathematics it's a good thing to focus on representation, transformation and constraint.

Take a look at this if you are interested.

 

[micromass]

The exercises in Spivak certainly have a reputation of being quite difficult. In fact, the exercises are more analysis exercises than calculus exercises.

If you want to be able to grasp calculus, then the level of exercises in Spivak is higher than what you need.

I suggest two books to you:
1) Schaum's outline of calculus: contains many exercises on a decent level, but not too difficult. If you can solve these exercises, then your calculus is ok.

2) Understanding Analysis by Abbott. This is a friendly introduction to analysis. It's maybe a bit nicer than Spivak and it'll help you tackling Spivak's exercises.

 

[mathwonk]

It is also possible you are not reading those books in enough depth. To understand apostol or spivak you must try to master every word in the section you read, and be able to reproduce every proof. The try the problems using ideas similar to those that have been used in the section read, plus more. you have to read actively, with pencil in hand, and probably writing out several pages for each page read.

 

[deRham]

I know the problem you're having, I think. It was the same with me when I first started reading mathematics. What you have to really endeavor to do is make those symbols on the page actually mean something to you (a picture, an idea, etc, etc).

However, solving problems of the sort you're probably dealing with involves TWO steps rather than just the one you're doing:

1) Learn the ideas, like I said.

2) Actually stare at the mathematical properties of the specific things they show you, and see HOW you can relate them back to the definitions by applying some mathematical trick. A simple example of this is that when you have a quotient of polynomials in one variable N, where N ranges over integers, you can read off the limit as N approaches infinity based on the highest degree terms of the polynomials.

There are two aspects to that example: one, you have to understand exactly what it means for that expression to approach a given limit. There's a baby step 1', where you say "well, it means that expression gets closer and closer to ___", which you learn in a basic calculus class. To complete step 1, you must know what that formally means: it means the mathematical difference between the expression and a given number called the limit can be made arbitrarily small as N gets larger.

But there's ANOTHER step, which is that you need to know how to actually show this mathematically, rather than just saying it intuitively. To translate the mathematical intuition that "the highest terms are what matter", try to mathematically change the entire expression so that the highest terms are all that are left. You do this by dividing numerator and denominator by something.

And by the way, you'll get better at noting the answers to step 2 as you just learn more math.

 

[eliya]

Everyone failed to mention this, but you should take a class that instructs how to prove things. Or at the very least get a book about it. I also got Spivak when I started Calculus, and while the theorems were the same as what I studied in Calculus, I couldn't understand the proofs.

I took a proofs class and I'm now taking Real Analysis and doing well. I don't think I could have done well without that introductory proof class, and I think I'd do even better if I had taken a logic and set theory class, which leads me to my next point.

A lot of people are fine with seeing a few basic ideas (like the kind of stuff you see in an intro to proofs class), and run with them. I, for instance, am not one of those. I like understanding things thoroughly, so the more detailed classes (and perhaps maybe the more rigorous ones?) work better for me. However, like I said before, taking a class about how to write and understand proofs was enough for my real analysis.

Bottom line is, you should get some idea of how to prove things. If you're looking for books, here's the book I used in my proofs class. I think it does a fantastic job explaining the ideas, but the problems aren't very challenging. They just help you understand how to set up a proof, but nothing more.
A really good book with good explanations is this one, which can be bought for way cheaper at other websites.
This book was recommended to me before, but I never got it because it's too expensive.
Lastly, this book is being recommended here a lot. I don't like the way it explains things, but it might work for you.

Also, what mathwonk said is really important. Reading mathematics is doing mathematics. You won't get a lot out of it if you don't sit and rewrite the proofs as you go along.
Also, the problems in those books are hard; you might spend a few hours on a problem until you figure out how to crack it. It gets better as you keep advancing though. These problems are also different than problems you've done before. With computational classes you usually start manipulating something until you get the results, and a lot of the time you do it on ``auto-pilot". Here you have to think a lot before you start manipulating things.

One last thing (promise). Analysis isn't hard. It just requires a lot of time. So be prepared to that. Also, don't stress out too much if you don't get to do analysis until you actually take a course in analysis. Take the classes you need to take to get to analysis (including a proofs class), and be prepared to work hard.

 

[brugel]

How to solve it by Polya or transition to advanced mathematics by douglas smith might help you learn how to do proves and how to start these problems. I went through Spivak after reading these and it was a lot of help. Also try making an outline and examples for each theorem.