Gary
GaryVasco
Posts: 3,352 
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Post by Gary on Dec 22, 2015 1:25:54 GMT
Vasco, Here are some comments on Ch 5, Ex. 5. (i) My copy of the book has “i sin x” where you have “i sin y”, so the part in parens. corresponds to $e^{ix}$. (ii) I wonder why Needham included such an easy problem. Perhaps to make the point that even very simple forms can be non-analytic. (iii) I think your solution is perfectly valid and probably the easiest, and I also considered that route, but I wonder if Needham intended this problem to provide an opportunity to use the polar-polar form of the C-R equations. I worked on that assumption. (I retract the comment---the Cartesian-Polar is more appropriate---but it was still interesting to try the polar-polar form.) (iv) The polar-polar form of the C-R equations seems the most natural to use for this exercise. What do you think? Gary By the way, the Preview button does not work for me. Attachments:nh.ch5.ex5.pdf (101.62 KB)
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Post by Admin on Dec 22, 2015 15:58:36 GMT
Gary
Part (i)
My copy of the book also has $i\sin x$. This was a typo in my solution, which is worked through after the typo as though the typo were not there! I have corrected it and republished.
Part (ii)
Not sure, maybe for the reasons you say.
Part (iii)
I agree with your final statement that Polar-Cartesian is more appropriate here.
Part (iv)
I agree that Polar-Polar seems more natural here. I'm not sure why I chose to convert and use Cart.-Cart. It's almost three years ago when I first did the exercise.
When you say that preview doesn't work, do you mean that the maths conversion doesn't work or that it doesn't work full stop?
It seems to work for me, but the preview itself doesn't look any different to the non-preview and the maths doesn't convert. I'll look into it.
Vasco
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Post by Admin on Dec 22, 2015 16:18:30 GMT
Gary
I have just tried Preview after emboldening some text and it does seem to work as far as this is concerned.
Vasco
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Post by Admin on Dec 22, 2015 18:41:19 GMT
Gary
Looking at your solution to exercise 5 again, here are some comments:
There is a typo in the heading where it says exercise 4 instead of exercise 5.
I've mentioned something similar to this before, but in most parts of the exercise you state the C-R equations as being satisfied, when the exercise is to show that they are. I think you should evaluate the two sides of a C-R equation and show that they are equal at the end of the proof, but that's just me.
In part (iii) I would avoid having to use the multifunction log and use the Polar-Cart form of the C-R, because it keeps things simple.
There is a typo in your solution to part (iv):
In your calculation of $\partial_{\theta}R$, on the second line you have an extra $e^{r\cos\theta}$ at the end
Vasco
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Dec 23, 2015 2:01:27 GMT
Gary Looking at your solution to exercise 5 again, here are some comments: There is a typo in the heading where it says exercise 4 instead of exercise 5. I've mentioned something similar to this before, but in most parts of the exercise you state the C-R equations as being satisfied, when the exercise is to show that they are. I think you should evaluate the two sides of a C-R equation and show that they are equal at the end of the proof, but that's just me. In part (iii) I would avoid having to use the multifunction log and use the Polar-Cart form of the C-R, because it keeps things simple. There is a typo in your solution to part (iv): In your calculation of $\partial_{\theta}R$, on the second line you have an extra $e^{r\cos\theta}$ at the end Vasco Vasco, Regarding the format of the exercises, take 5 (ii) as an example, I stated the Cauchy-Riemann equations and then I evaluated both sides of the first C-R equation. They are obviously not equal, so I stated that they are not satisfied. I think I have followed that format most of the time. Do you mean that at line 7, it would be better form to insert the following before stating that the equations are not satisfied? $\partial_x u = -sin x$ and $ \partial_y v = cos y$, so $\partial_x u \neq \partial_y v$ I can do that. Regarding part (iii), I agree. Regarding part (iv), fixed and I will replace the attachment. Regarding the preview, absolutely nothing happens when I click the Preview button, except that the bottom of my window has "javascript: void(0);", so perhaps there is a problem with the javascript on my computer. Thanks. The holidays are upon us, so I may be slow to respond. Gary |
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Post by Admin on Dec 24, 2015 7:53:04 GMT
Gary Looking at your solution to exercise 5 again, here are some comments: There is a typo in the heading where it says exercise 4 instead of exercise 5. I've mentioned something similar to this before, but in most parts of the exercise you state the C-R equations as being satisfied, when the exercise is to show that they are. I think you should evaluate the two sides of a C-R equation and show that they are equal at the end of the proof, but that's just me. In part (iii) I would avoid having to use the multifunction log and use the Polar-Cart form of the C-R, because it keeps things simple. There is a typo in your solution to part (iv): In your calculation of $\partial_{\theta}R$, on the second line you have an extra $e^{r\cos\theta}$ at the end Vasco Vasco, Regarding the format of the exercises, take 5 (ii) as an example, I stated the Cauchy-Riemann equations and then I evaluated both sides of the first C-R equation. They are obviously not equal, so I stated that they are not satisfied. I think I have followed that format most of the time. Do you mean that at line 7, it would be better form to insert the following before stating that the equations are not satisfied? $\partial_x u = -sin x$ and $ \partial_y v = cos y$, so $\partial_x u \neq \partial_y v$ I can do that. Regarding part (iii), I agree. Regarding part (iv), fixed and I will replace the attachment. Regarding the preview, absolutely nothing happens when I click the Preview button, except that the bottom of my window has "javascript: void(0);", so perhaps there is a problem with the javascript on my computer. Thanks. The holidays are upon us, so I may be slow to respond. Gary Gary I'm only being pernickety, but what you suggest would be good, and maybe remove the line "C-R: $\partial_xu=\partial_yv$ as it could confuse because it says that the C-R are satisfied, which turns out to be incorrect in this case. I think the fact that you say "Apply the Cauchy-Riemann test" is sufficient, but it's only a matter of style. I've done some experiments with preview and when I start a new post if I click on BBCode I see a change and then preview takes me back. I think when you start a new post you are by default already in Preview mode. If you edit you are automatically in BBCode mode until you click Preview. I get the same javascript message as you, so I don't think there's a problem with your system. Try some experiments by clicking on BBCode and see what happens. I also could be a bit distracted over the holiday period. Vasco |
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Dec 24, 2015 18:47:04 GMT
Vasco,
I see now. Even though they are easily memorized, I like to have the test conditions in front of me when I begin, but perhaps it would be more clear if I wrote "To satisfy the Cauchy-Riemann test, the following must be true: ∂xu=∂yv and ∂xv=-∂yu." Or I could delete the prologue as you suggest. I am just going through these exercises to learn, but I don't want to confuse anybody either, and format is part of the exercise. I will let the situation be my guide and keep your suggestions in mind.
Gary
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