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Chapter 10  Problem with Xi http://www.roadtoreality.info/viewtopic.php?f=22&t=1978 
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Author:  fruity [ 31 Jul 2012, 18:55 ] 
Post subject:  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 '1form'. I think, I miss anything to understand this section. Does anyone know how it works? Greetings, fruity 
Author:  vasco [ 31 Jul 2012, 20:31 ] 
Post subject:  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 
Author:  vasco [ 31 Jul 2012, 20:39 ] 
Post subject:  Re: Chapter 10  Problem with Xi 
here is an operator, so 
Author:  vasco [ 31 Jul 2012, 20:58 ] 
Post subject:  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. 
Author:  fruity [ 31 Jul 2012, 22:31 ] 
Post subject:  Re: Chapter 10  Problem with Xi 
Thanks... I will have to think about it (,too). 
Author:  fruity [ 01 Aug 2012, 07:47 ] 
Post subject:  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? 
Author:  vasco [ 01 Aug 2012, 11:50 ] 
Post subject:  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. 
Author:  fruity [ 01 Aug 2012, 18:18 ] 
Post subject:  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 ? 
Author:  vasco [ 02 Aug 2012, 07:09 ] 
Post subject:  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. 
Author:  fruity [ 02 Aug 2012, 09:12 ] 
Post subject:  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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