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Post by Admin on Jan 26, 2023 9:29:25 GMT
Mondo
From near the bottom of page 237 we have
$\displaystyle\sigma=\text{Im}\bigg[\frac{f''(p)\xi}{f'(p)}\bigg]$
So I disagree with your result for $\sigma$.
As for your other points, I think you are confusing mappings and derivatives, which is easy to do,
We can combine as many mappings as we like and write things like
$w=f(g(h(z)))$ where $f,g,h$ are all mappings of $z$, each one being completely independent of the others.
We can also differentiate any function $f(z)$ any number of times: $f',f'',f''',......$ at the same point.
Don't get confused by the fact that the complex mappings, when applied to infinitesimal vectors, just happen to use the amplitwist concept.
I would suggest reading the last paragraph on page 198.
Vasco
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Post by mondo on Jan 27, 2023 6:55:10 GMT
Mondo From near the bottom of page 237 we have $\displaystyle\sigma=\text{Im}\bigg[\frac{f''(p)\xi}{f'(p)}\bigg]$ So I disagree with your result for $\sigma$. Yes I also see it is wrong now. $(f'(q) - f'(p)) = \chi$ and not $\sigma$ as I wrote in my previous post. That this is wrong can be easily seen by imagining $f'(q)$ at the imaginary axis and $f'(p)$ on the real axis, then $\sigma = 90$ but $ang(\chi)$ is clearly not a right angle. However $arg(f'(p)) - arg(f'(q))$ must give us $\sigma$ as can be seen from the right hand side of figure [18]. It is just the angle between these two vectors. I also don't think this is correct $\sigma = arg(f''(\tilde{p}))$ as well as calculations below: To do the amplitwist of an amplitwist we could apply the function $f'$ to $\widetilde{p}$ and write $\mu=f''(\widetilde{p})\widetilde{\xi}$ Since $\widetilde{\xi}=f'(p)\xi$, we can write this as $\mu=f''(\widetilde{p})f'(p)\xi$. So the total twist applied to $\xi$ is $\arg{[f''(\widetilde{p})]}+\arg{[f'(p)]}$, where $\widetilde{p}=f(p)$. This is because $\mu=f''(\widetilde{p})\widetilde{\xi}$ is some unknown point, neither $f'(p)$ nor $f'(q)$ which we need for $\sigma$. $f''(\widetilde{p})\widetilde{\xi} = f'''(p)\xi$ - that is too much differentiation. I would conclude here that the only ways to get that angle are what author did and what I wrote above as a difference between angles/twists of function derivatives. |
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Post by Admin on Jan 27, 2023 8:23:32 GMT
Mondo
OK. Those were just meant to be examples.
Vasco
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Post by mondo on Jan 28, 2023 23:59:35 GMT
Thank you Vasco for the help with this subsection - it wasn't easy to get through it. Even though this section is optional I think figure [18] really helps to understand the whole amplitwist concept.
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Post by Admin on Jan 29, 2023 8:20:14 GMT
Thank you Vasco for the help with this subsection - it wasn't easy to get through it. Even though this section is optional I think figure [18] really helps to understand the whole amplitwist concept. Mondo No problem. I remember it was difficult when I first studied it about 5 years ago. But as you say figure 18 is a great help in making sure you understand the amplitwist concept, which is what the book is all about. It is the amplitwist concept which allows Needham to develop the subject in a visual way, which is what the book is all about. Vasco |
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