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Archived: 07 Aug 2014, 09:32
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Chapter 10 - Problem with Xi
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fruity
Joined: 31 Jul 2012, 17:53 Posts: 5
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 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
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31 Jul 2012, 18:55 |
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vasco
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Joined: 07 Jun 2008, 08:21 Posts: 235
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 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
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31 Jul 2012, 20:31 |
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vasco
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Joined: 07 Jun 2008, 08:21 Posts: 235
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 Re: Chapter 10 - Problem with Xi
 here is an operator, so ![\xi F(x,y)=[A \frac{\partial }{\partial X} + B \frac{\partial }{\partial Y}]F=A \frac{\partial F}{\partial X} + B \frac{\partial F}{\partial Y} \xi F(x,y)=[A \frac{\partial }{\partial X} + B \frac{\partial }{\partial Y}]F=A \frac{\partial F}{\partial X} + B \frac{\partial F}{\partial Y}](latexrender/pictures/14f9e88a36d3347049750d33dbc6d337.png)
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31 Jul 2012, 20:39 |
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vasco
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Joined: 07 Jun 2008, 08:21 Posts: 235
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 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.
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31 Jul 2012, 20:58 |
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fruity
Joined: 31 Jul 2012, 17:53 Posts: 5
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 Re: Chapter 10 - Problem with Xi
Thanks... I will have to think about it (,too).
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31 Jul 2012, 22:31 |
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fruity
Joined: 31 Jul 2012, 17:53 Posts: 5
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 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 ![\xi F(x,y)=[A \frac{\partial }{\partial X} + B \frac{\partial }{\partial Y}]F=A \frac{\partial F}{\partial X} + B \frac{\partial F}{\partial Y} \xi F(x,y)=[A \frac{\partial }{\partial X} + B \frac{\partial }{\partial Y}]F=A \frac{\partial F}{\partial X} + B \frac{\partial F}{\partial Y}](latexrender/pictures/14f9e88a36d3347049750d33dbc6d337.png) 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?
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01 Aug 2012, 07:47 |
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vasco
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Joined: 07 Jun 2008, 08:21 Posts: 235
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 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.
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01 Aug 2012, 11:50 |
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fruity
Joined: 31 Jul 2012, 17:53 Posts: 5
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 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  ?
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01 Aug 2012, 18:18 |
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vasco
Supporter
Joined: 07 Jun 2008, 08:21 Posts: 235
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 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.
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02 Aug 2012, 07:09 |
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fruity
Joined: 31 Jul 2012, 17:53 Posts: 5
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 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.
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02 Aug 2012, 09:12 |
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