Contour Integration: Understanding R. Penrose's The Road To Reality

[springo]

Hi,
I've started reading The Road To Reality, by R. Penrose, and I've made my way through the first six chapters. Now the author talks about contour integration, and I'm really confused with his explanation.
He says a few things about homologous and homotopic deformations, and then, he talks about the contour integral of $$\frac{1}{z}$$, and finally Cauchy formula.
Do you know anywhere I could possibly find more (comprehensible) information on those topics?
Thank you very much.

 

[Gib Z]

How much math do you know? And do want the mathematical or physical viewpoint, they'll lead you a bit differently. Wikipedia might help for now.

 

[Swapnil]

Try these. Hope it helps you better.

http://www.maths.abdn.ac.uk/courses/mx3522/site/notes.html

http://www.math.gatech.edu/~cain/winter99/

 

[springo]

How much math do you know? And do want the mathematical or physical viewpoint, they'll lead you a bit differently. Wikipedia might help for now.

All the calculus I know comes from what I read in that book, because I'm an 11th grade student. I've also read about logarithms (multiple valuedness, e...), complex numbers (powers, complex plane...) and I finished the chapter about real-number calculus (smoothness, analytic functions...).
For now, I'd like to understand the concept matematically, rather than the physical implications.

PS: Thanks for those links, Swapnil, I'm trying to read that right now.

 

[Gib Z]

Well, unless that book is a textbook, I am not sure how you learned calculus from that. Try a calculus textbook.

 

[springo]

Well, unless that book is a textbook, I am not sure how you learned calculus from that. Try a calculus textbook.

OK, I'll read a textbook before going further in my book :)
Thanks.

 

[Dbjergaard]

I just ordered road to reality, I'm apprehensive about starting it... I taught myself differential calculus this summer using a text and "Calculus Made Easy" (which I would suggest as a supplement to the text) Anyway, I would really like to know how rigorous Road to Reality is... Is it a large volume of unproven tricks to do math in physics problems, or are the explanations laid out in a clear and proven manner? My book will be here next wednesday.

 

[springo]

From what I've read, I can tell you that it is difficult to understand everything Penrose explains, unless you have previous knowledge of what he is talking about.
He gives you some exercises, which have three levels of difficulty. If you solve the 'easy' ones, it means that you understood what he is talking about; 'medium' level ones usually require you to know more than what he said, and 'difficult' are usually impossible unless you know much more math/physics than what you've just read.
These exercises are usually asking you to prove the assertions he makes. You should be able to find the solutions here, but he has only published the answers for the first three chapters (out of thirty-something).
I mean, the book is fine, but (I'm just realizing) it requires you to do some extra work if you really want to understand what you read.

 

[mathwonk]

i first began to appreciate this subject from the discussion of complex logarithms in courant.

the idea is that you want the log of z to be the integral of dz/z takken from 1 to z. But you know the value of the log depends on some choices, namely iof the argument of the log.

so the explanation is that you have to choose a path of integration from 1 to z and niot pasing through 0. then the resukt is that two different paths give the same integral if and only if the close path formed by going forward along one patha nd backward along the other, is a closed path not winding at all around 0.

so the best result is that the integral of a differnetial f(z)dz where f is holomorphic, is the same for two different paths, provided the closed path they form together as above, does not wind around any point outside the domain of f.

 

[mathwonk]

i have never read road to reality but my feeling from reading posts here is that most people trying to read it do not have the mathematical prerequisites.

 

[Dbjergaard]

and those requisites would be...
I have calculus through integration (we just finished transcendental integrals)
I also have a firm foundation in physics, mainly in mechanics

 

[mathwonk]

well i don't know since i haven't read it, but i get the second hand impression it uses a lot of math. almost like it was aimed at professionals.

 

[DeadWolfe]

The book is intended to be accessible to people who have no math understanding whatsoever, and it goes from basis calculus to fibre bundles and gauge connections.

Needless to say, it fails miserably in being accessible to people who don't know most of the material already, though it certainly is an interesting read.

 

[springo]

DeadWolfe is right, I mean, in the introduction, he explains how to add and multiply fractions, but in chapter 5 (less than 100 pages later), he is already talking about complex logarithms.

 

[Gib Z]

He certainty goes very quickly, but you may realize the physical content of his book requires the mathematics, and if he chose the mathematics could have been much more comprehensive. The reason he chose not to was because he *attempted* to make it accessible to the layman. Not always possible for a layman to understand that kind of physics/math, no matter how hard the author tries.

 

[matt grime]

DeadWolfe is right, I mean, in the introduction, he explains how to add and multiply fractions, but in chapter 5 (less than 100 pages later), he is already talking about complex logarithms.

He takes 100 pages to get there? It's easily doable in under 2. I would imagine it easy to produce a rigorous treatment starting from fractions to the principal argument of log in less than two pages, developing all you need to know, which is not to say all you ought to know. (Depends on how much you want to prove about the real numbers - just go for the space obtained by allowing infinite sums of rationals). A steep learning curve is no barrier (he said somewhat optimistically, if not hypocritically).

 

[springo]

Well, in his way, he also talks about stuff like hyperbolic geometry (chapter 1), which seems complicated (to me). However I believe that this amount of information (about which I hadn't heard of before) is what makes the book interesting.