The Road to Reality http://www.roadtoreality.info/ 

Exercise [08.04] c http://www.roadtoreality.info/viewtopic.php?f=19&t=279 
Page 1 of 1 
Author:  vasco [ 28 Feb 2009, 09:01 ] 
Post subject:  Exercise [08.04] c 
Here is a development of the previous proof, which takes into account the special case when the circle in the zplane passes through the origin and is transformed into a straight line (circle with infinite radius as Penrose calls it). I noticed a small typing error in the attached .pdf file which I have now corrected. "Equation (5) is of the same form as equation (1)" should be "Equation (5) is of the same form as equation (2)" Thanks to Azrael's post below I have corrected a typo and added a clarification. See Azrael's post for details. The pdf file below has also been amemded to correct the typo and add the clarification. Edited by Vasco on 10/10/2011. Vasco Attachment: 
Author:  cwjian [ 25 Jul 2011, 18:13 ] 
Post subject:  Re: Exercise [08.04] c 
Vasco, I have a question. Why does shifting the plane by i upwards not work, whereas rotating the plane by pi/2 radians and then subtracting 1 does? 
Author:  vasco [ 25 Jul 2011, 21:32 ] 
Post subject:  Re: Exercise [08.04] c 
Hi I am a bit confused! Are you sure you are talking about this exercise? Don't you mean Exercise 8.7? If you do mean this exercise, can you explain in a bit more detail what you mean? Thanks Vasco 
Author:  cwjian [ 28 Jul 2011, 15:22 ] 
Post subject:  Re: Exercise [08.04] c 
Sorry Vasco, yes, I posted in the wrong thread... Would you mind if I continued this discussion in the other thread? Or should I continue it here? 
Author:  vasco [ 28 Jul 2011, 15:51 ] 
Post subject:  Re: Exercise [08.04] c 
Hi cwjian I think it would be more useful to everybody if you transferred to the other thread. Presumably the one for exercise 8.7  Is that right? Also could you explain in a bit more detail what it is that you do not understand? Thanks Vasco 
Author:  cwjian [ 29 Jul 2011, 18:19 ] 
Post subject:  Re: Exercise [08.04] c 
Hi Vasco, I've posted in the Problem 8.07 thread. cwjian 
Author:  Azrael84 [ 09 Oct 2011, 15:05 ] 
Post subject:  Re: Exercise [08.04] c 
Nice solution. I believe there are few typos, equation (1) should really look like: z=z_0+re^{it}, but this doesn't affect the proof obviously. Also on the last page I think you mean to take z_0=x_0+iy_0 (missing an i as z_0 is complex constant), again just some typos hopefully make it easier for those going through your proof. 
Author:  vasco [ 10 Oct 2011, 07:48 ] 
Post subject:  Re: Exercise [08.04] c 
Azrael84 wrote: Nice solution. I believe there are few typos, equation (1) should really look like: z=z_0+re^{it}, but this doesn't affect the proof obviously. Also on the last page I think you mean to take z_0=x_0+iy_0 (missing an i as z_0 is complex constant), again just some typos hopefully make it easier for those going through your proof. Thanks for reading my solution. I'm glad you like it! I agree that on the last page it should be z_0=x_0+iy_0, but I don't agree with your first point. r is a complex number here and so z=z_0+r is correct. That's why I called it a radius vector in line 1. I think that I should have added r=constant to make it clearer. As z changes so does r, but only the argument of r changes not its modulus. I will amend my first post above and upload a corrected and amended pdf file. 
Author:  Azrael84 [ 10 Oct 2011, 19:01 ] 
Post subject:  Re: Exercise [08.04] c 
vasco wrote: but I don't agree with your first point. r is a complex number here and so z=z_0+r is correct. That's why I called it a radius vector in line 1. I think that I should have added r=constant to make it clearer. As z changes so does r, but only the argument of r changes not its modulus. I will amend my first post above and upload a corrected and amended pdf file. Ah, OK. Yes I agree it makes sense if r is complex. I thought you had in mind 'r' as the radial type parameter of polar coords centered about the point z_0 and had accidentally missed off the e^{it} factor... 
Page 1 of 1  Archived: 07 Aug 2014 
Powered by phpBB © 2000, 2002, 2005, 2007 phpBB Group http://www.phpbb.com/ 