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Post by Admin on Jul 22, 2023 15:28:05 GMT
Mondo
Here's hint to the answer to your question in reply #42: Which angles is Needham referring to on page 190?
Vasco
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Post by mondo on Jul 22, 2023 18:24:05 GMT
Thank you Vasco. I think he is talking about the angle between the real axis and a particular complex number in a polar form. But as I said the angle is always doubled after the mapping of $z^2$. This sentence is also confusing because you can take it as $0^2$ doubles angles but it doesn't as it is still $0$.
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Post by mondo on Jul 22, 2023 18:37:14 GMT
$(e^{i(\theta + h)})^2 = e^{2i(\theta + h)}$ so no matter if our number was at $0$ angle before we did the mapping or some other angle, it is always at least doubled.
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Post by Admin on Jul 22, 2023 18:39:37 GMT
Mondo
The way I see it, he is saying that at any point in the left diagram of [1]. the angle between any two lines passing through the point will be the same on the right hand diagram at the corresponding point, except at zero and infinity where the angle between lines passing through these points is doubled when we go from left to right.
In the diagram the lines are drawn at right angles because Needham wants to show later that squares are mapped to squares
Vasco
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Post by mondo on Jul 22, 2023 18:52:13 GMT
OK, that is a good point. From the figure [1] we can also see that this behavior is persistent up until we hit the rim of the unit circle. How to show is algebraically? This means that if I square $e^{i\theta}$ the angles will be doubled but if I do the same for $2e^{i\theta}$ they won't but, we can quickly verify it is not true as $(2e^{i\theta})^2 = 4e^{2i\theta}$. I guess here we again think about infinitesimal increments but again for $e^{i\theta}$ no matter how small I increment $\theta$ it is always doubled after the $z^2$ mapping.
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Post by mondo on Jul 22, 2023 19:07:10 GMT
Even if I write it from derivative definition $\frac{(e^{i(\theta + h)})^2 - (e^{i\theta})^2}{h} = \frac{e^{2i\theta} + e^{2ih} - e^{2i\theta}}{h} = \frac{e^{2ih}}{h}$
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Post by Admin on Jul 22, 2023 19:12:27 GMT
Mondo
The point is that all curves through a given point are rotated through the same angle by the mapping (but you don't accept this do you?) and so the angle between them stays the same. It's very simple if you accept the idea of the amplitwist.
Vasco
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Post by mondo on Jul 22, 2023 19:21:11 GMT
I think it is true but I would love to prove it algebraically, so far I can't. You brought a good point about the figure [1]. It is interesting that angles are only doubled as long as we operate within a unit circle, Vasco do you know how to show is algebraically?
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Post by Admin on Jul 22, 2023 19:29:31 GMT
Mondo
No it all works inside the unit circle as well as outside.
Vasco
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Post by Admin on Jul 22, 2023 19:32:51 GMT
Mondo
It's shown algebraically on page 198!!!!
Vasco
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Post by Admin on Jul 22, 2023 19:40:16 GMT
Mondo
And elsewhere in chapter 4 and chapter 5 is crammed full of examples.
Vasco
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Post by mondo on Jul 22, 2023 20:42:46 GMT
Mondo It's shown algebraically on page 198!!!! Vasco No it's not what I wanted. It just shows how to represent a derivative as the "amplitwist". While I want to get an algebraic feel of the fact that angles are not doubled beside the origin. This is how I would show this: let's take two complex numbers $A=e^{i\theta}$ and $B=e^{i(\theta+h})$ so $B$ is very close to $A$ on the unit circle. Now, let's apply the complex mapping $z^2$, our point A is mapped to $e^{2i\theta}$ while $B = e^{2i(\theta + h)}$ and now a crucial observation in respect to the origin $z=0$ angles of both are doubled, but in respect to each other they are only $2h$ apart! Hence the angles between them is definitely not doubled. So to summarize the doubling of angle that author is talking about in the first paragraph of page 190 is in regards to the origin. The phrase he uses "except at z=0" is confusing a bit. However it is much harder to think about this behavior at the $\infty$, do you have some clue Vasco? Especially that there is no infinity in rotation - infinity ends every $2\pi$ |
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Post by Admin on Jul 22, 2023 20:53:23 GMT
Mondo
Have you read the last paragraph on page 189? Angles in this context are angles between intersecting lines such as those in figure 1
The concept of conformal is first introduced on pages 130-131.
Vasco
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Post by mondo on Jul 22, 2023 21:01:33 GMT
I will reread it again. Especially page 130-131 as you suggest.
Vasco, can you also give me a hint as to why my attempt to calculate the derivative of $z$ in polar form WRT length and angle separately fails to match the derivative WRT to $z$? For $(z^2)' = 2z$ but when I do it for $(|z|^2e^{2i\theta})'$ it is not quite the same - vector length matches but twist doesn't. $d/d\theta [e^{2i\theta}] = 2ie^{2i\theta}$
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Post by Admin on Jul 22, 2023 21:51:09 GMT
I will reread it again. Especially page 130-131 as you suggest. Vasco, can you also give me a hint as to why my attempt to calculate the derivative of $z$ in polar form WRT length and angle separately fails to match the derivative WRT to $z$? For $(z^2)' = 2z$ but when I do it for $(|z|^2e^{2i\theta})'$ it is not quite the same - vector length matches but twist doesn't. $d/d\theta [e^{2i\theta}] = 2ie^{2i\theta}$ Mondo Do you mean derivative of $z^2$ with respect to $z$? Vaasco |
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