How do you study real analysis?

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  • #1
cks
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i'm taking the course of real analysis this semester. However, I find difficulty in understanding it. there're so many terminology and terms. the textbook i'm using is "principles of mathematical analysis" . Sometimes, I read and read to understand a definition or theorem, but still fails to understand it.

how do you overcome the difficulty in learning it?
 

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  • #2
HallsofIvy
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The first thing you need to do is learn the definitions very well. Then look at some of the theorems proved early in the book. Particularly observe how the specific words of the definitions are used in the proofs.
 
  • #3
Gib Z
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I am sure I will be brandished for saying this but if you are just looking to pass through your tests fine, there are things where learning the theorems and applying them works just fine. Like, Circle Geometry tests. However its always better, and required, even if you just want to pass your tests, for real analysis, to understand the definitions of all the terms, operations, and exactly where and how the theorems can be used. Try to learn proofs to most of the important theorems.

And for everything, try to understand them in several ways. Thats a tip i read from Terry Tao's blog. Ie For all theorems and definitions, know what they mean algebraically, geometrically, intuitively etc etc. EG The derivative geometrically is the slope of the tangent at a point. Algebraically is it the limit [tex]\lim_{h\to 0} \frac{ f(x+h) - f(x)}{h}[/tex]. Intuitively it is the instantaneous rate of change of a function of one variable with respect to another.

Or the mean value theorem for integrals: Geometrically it is the area of the "mean value" rectangle. Algebraically it is [tex] \frac{1}{b-a} \int^b_a f(x) dx[/tex]. Intuitively it is the average value of the function between the bounds b and a.

You get the point I'm sure.
 
  • #4
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Hello i too am taking this module for this semester and these proofs really are tricky at times. they are simple true but to think of what to do :S

So what other advice can i get from u guys?(and gals)
 
  • #5
Gib Z
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Learn almost every single proof for theorems, but don't remember them step by step, you'll understand the proof better if you just remember the critical parts of the proof. Sometimes you'll only need the name of a specific method :) Like for Euler's Formula, use Taylor Series. Or the circle geometry theorem, known to some as the "Power of a point" theorem, but I just remembered the statement: The product of the intercepts of 2 chords are equal. For that, just remember to prove Congruency.
 
  • #6
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only the important steps? i see
but i havent done euler's formula yet :S
maybe later. uni started only 2weeks back
 
  • #7
Gib Z
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O well, even if you don't understand what it's implications, you could be familiar with the functions involved: [tex]e^{ix} = \cos x + i \sin x[/tex]. Anyway, Only the important steps!
 
  • #8
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i know this formula done it last year in my A level
yeah thx a lot, if i get any problem understanding i'll aks :)
 
  • #9
Memorize all definitions and theorems if possible. That should be your first goal. Next, try to work through the proofs YOURSELF, only looking at a line from the text if you get stuck (not like 1 min stuck, like 30 min stuck!). Working through the text like that. Then last but not least try some homework questions, think about examples/counterexamples, draw pictures to illustrate theorems!
 
  • #10
mathwonk
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get a more explanatory book, like one by george simmons.
 
  • #11
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Don't let it get too abstract. For each definition/theorem come up with the simplest, (but non-trivial) concrete example, and make sure you understand why the definition/theorem makes sense.

Then try to come up with the most likely counterexamples to the theorem (or definition), and convince yourself that those counterexamples fail.

I read thru Munkres' Topology before Rudin's Priciples of Mathematical Analysis. Munkres does have plenty of good examples, and makes a good companion text.
 
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  • #12
cks
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Yes, I get your point. It's very motivating. Thanks.



I am sure I will be brandished for saying this but if you are just looking to pass through your tests fine, there are things where learning the theorems and applying them works just fine. Like, Circle Geometry tests. However its always better, and required, even if you just want to pass your tests, for real analysis, to understand the definitions of all the terms, operations, and exactly where and how the theorems can be used. Try to learn proofs to most of the important theorems.

And for everything, try to understand them in several ways. Thats a tip i read from Terry Tao's blog. Ie For all theorems and definitions, know what they mean algebraically, geometrically, intuitively etc etc. EG The derivative geometrically is the slope of the tangent at a point. Algebraically is it the limit [tex]\lim_{h\to 0} \frac{ f(x+h) - f(x)}{h}[/tex]. Intuitively it is the instantaneous rate of change of a function of one variable with respect to another.

Or the mean value theorem for integrals: Geometrically it is the area of the "mean value" rectangle. Algebraically it is [tex] \frac{1}{b-a} \int^b_a f(x) dx[/tex]. Intuitively it is the average value of the function between the bounds b and a.

You get the point I'm sure.
 
  • #13
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Have you taken a course in Analysis? Normally they have an 'introductory' Analysis course and then you can take Real Analysis and Functional Analysis etc.
 
  • #14
cks
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get a more explanatory book, like one by george simmons.

I hvae borrowed a book of his. there are a lot of explanations in the first few chapters, and clarify a lot of doubts. thanks
 
  • #15
cks
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The title of the book by George F. Simmons is

Introduction to Topology and Modern Analysis
 
  • #16
cks
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To #13

I major in physics but am taking this course because I believe it can provides me a solid foundation for my future research in Quantum Mechanics.
 
  • #17
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Im majoring in maths with computer and i dont have introductory analysis(i guess) coz the course is real analysis.
Ok will look for the book u guys recommended
 
  • #18
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To #13

I major in physics but am taking this course because I believe it can provides me a solid foundation for my future research in Quantum Mechanics.
You should take an introductory course in analysis. Frankly, the real analysis course is going to be very demanding and will expect you to be at a certain level of mathematical competency.
 
  • #19
For introductory texts I'd recommend Munkres Topology for your topology part, and Marsden's Elementary Classical Analysis. It's easier than the texts you should have a look at after you do Marsden (Rudin's Principles of Analysis or Pugh's Real Mathematical Analysis), but not stupid like some intro texts in that it does things topologically (which is how it should be done from the start IMO). Also, it has worked examples with proofs; very useful for a first course.
 
  • #20
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I second the Pugh recommendation. It covers intro real analysis in a clearer way than Rudin.
 
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  • #21
Haelfix
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Analysis at the upper undergrad or early grad lvl is tricky in the sense that a lot of the things you see, you already know from calculus or some other course and have an intuition for. So there is a temptation to really be lax in your responsibilities as a student. The tricky part is in the rigor thats required to actually prove things, as you realize a lot of the old proofs you remembered are actually quite innaccurate and not general (they tend to miss pathologies and all sorts of extra interesting and important structure).

The hard part is both the great strength and great weakness of Rudin. Just using his text, you are forced to memorize and learn the gritty details of the proofs ad nauseum, but at the same time you can fall a bit into 'rigor' mortis and you forget about the intuition.

That was the point roughly in my career where I decided I wanted to be a physicist and not a mathematician. I stuck with it for a few more years, but that was the turning point I think.
 
  • #22
cks
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Thank you to all you guys, I will put all my efforts in it.
 
  • #23
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O well, even if you don't understand what it's implications, you could be familiar with the functions involved: [tex]e^{ix} = \cos x + i \sin x[/tex]. Anyway, Only the important steps!

I often see this formula, [tex]e^{ix} = \cos x + i \sin x[/tex]
but I forgot its meaning.Can sb tell me is it
only a definition or can be proven to be so?
 
  • #24
Gib Z
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Someone on the forums, please tell me what the slash though uiulic's name means? If it means he isn't coming back, the rest of this post is useless.

That formula can be proven by letting ix be the variable in the maclaurin series for e^x, and showing that it is the same as the maclaurin series for cos x + i sin x.
 

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