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Chapter 10 - Problem with Xi
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Joined: 31 Jul 2012, 17:53
Posts: 5
Chapter 10 - Problem with Xi
Hi,
At page 187, is introduced:

But I'm not sure what is meant by this.
Somewhere is written that but I simply don't understand this section. For example, doesn't return a vector, right? Why is then called a 'vector field'?
Furthermore, I still wonder what is 'dx' and the '1-form'. I think, I miss anything to understand this section.
Does anyone know how it works?
Greetings,
fruity

31 Jul 2012, 18:55
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Joined: 07 Jun 2008, 08:21
Posts: 235
Re: Chapter 10 - Problem with Xi
You may or may not find it useful to read this discussion in the forum:
viewtopic.php?f=4&t=1439

31 Jul 2012, 20:31
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Joined: 07 Jun 2008, 08:21
Posts: 235
Re: Chapter 10 - Problem with Xi
here is an operator, so

31 Jul 2012, 20:39
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Joined: 07 Jun 2008, 08:21
Posts: 235
Re: Chapter 10 - Problem with Xi
can be thought of as a vector since it is the derivative in the direction of the x axis.

31 Jul 2012, 20:58

Joined: 31 Jul 2012, 17:53
Posts: 5
Re: Chapter 10 - Problem with Xi
Thanks... I will have to think about it (,too).

31 Jul 2012, 22:31

Joined: 31 Jul 2012, 17:53
Posts: 5
Re: Chapter 10 - Problem with Xi
Ok, I what I've understood now:
is the direction we want to calculate the slope in.
is the covector to the vector along the contour, but I'm not sure. There is the equation , which says me that the slope vanishes when the vector points in the right direction (along the contour). also says me that has to be the covector of , right? I think I still miss something related to ...

vasco wrote:
here is an operator, so

So this equation in the book didn't mean the vector but the operator or , which returns a number...
Furthermore, is called a vector field. Is now an operator or a vector or a vector field?

01 Aug 2012, 07:47
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Joined: 07 Jun 2008, 08:21
Posts: 235
Re: Chapter 10 - Problem with Xi
fruity wrote:
Thanks... I will have to think about it (,too).
fruity wrote:
Ok, I what I've understood now:
is the direction we want to calculate the slope in.
is the covector to the vector along the contour, but I'm not sure. There is the equation , which says me that the slope vanishes when the vector points in the right direction (along the contour). also says me that has to be the covector of , right? I think I still miss something related to ...

is not a vector, but a scalar field on S (see page 183 section 10.2 near the top), so you can't write .
is also not a vector.

01 Aug 2012, 11:50

Joined: 31 Jul 2012, 17:53
Posts: 5
Re: Chapter 10 - Problem with Xi
Sorry, I meant

and

. What I still don't understand is why should "give the direction of the contours" (Fig. 10.8). Isn't it the same as ?

01 Aug 2012, 18:18
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Joined: 07 Jun 2008, 08:21
Posts: 235
Re: Chapter 10 - Problem with Xi
Hi fruity
By the way I don't understand this topic either, I'm just hoping that by discussing it with someone else I might get there.

02 Aug 2012, 07:09

Joined: 31 Jul 2012, 17:53
Posts: 5
Re: Chapter 10 - Problem with Xi
Thanks for your help. I think what I know now is enough for reading the next chapters...
Maybe the use of these things in later chapters will help, too.

02 Aug 2012, 09:12
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