![]() | This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 |
Shouldn't the vectors in the picture for vector field on a sphere be tangent vectors? —Preceding unsigned comment added by 134.226.81.3 ( talk) 14:45, 21 April 2009 (UTC)
The introduction is rather good as is. However, I think it would be important to draw an analogy, as the reader can have a picture in his mind before attacking the maths. I'm sure my 14 year-old self would have liked it.
If there isn't any disapproval, I'd like to quote the own article and expand the second sentence to:
Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point. They can be compared to scalar fields, which simply associate a number or scalar to every point in space, such as the temperature in a room.
Also, my bogus-o-meter bounces when reading "a vector field [...] associates a vector to every point in [...] Euclidean space." I can sure imagine a vector field in non-Euclidian space... Thoughts? FelisSchrödingeris ( talk) 00:58, 17 January 2008 (UTC)
There seems to be a mistake in the relation given in Section 1.1 for translating the expression of a vector field from one coordinate system to another. I think it should read:
.
This issue has been discussed in a Usenet thread. Could someone please verify this? Thanks. -- Pouya Tafti 20:56, 30 November 2006 (UTC)
This page needs an explanation of vector fields acting as operators on functions since this concept is used elsewhere e.g. Lie algebra - Gauge 04:02, 3 Aug 2004 (UTC)
If you think a page should be expanded on then be bold and do it yourself. There is no need to discuss it on the talk page. On the other hand if you are completely restructering a page a short discussion on the talk page might be apropriate to prevent an edit war. MathMartin 10:08, 3 Aug 2004 (UTC)
This stuff is at Tangent_vector#Tangent_vectors_as_directional_derivatives. The problem is that we have too many pages with unclear relations to eachother. -- MarSch 17:31, 11 Jun 2005 (UTC)
This article is part of a series of closely related articles for which I would like to clarify the interrelations. Please contribute your ideas at Wikipedia talk:WikiProject Mathematics/related articles. -- MarSch 14:05, 12 Jun 2005 (UTC)
The following paragraph is now in the article:
The difference between a scalar and a vector field is in how their coordinates respond to coordinate transformations. Coordinates of scalars, by definition, don't transform at all. See example for Euclidean and cylindrical coordinates:
I beg to contradict you. A vector is a quantity with n components, an element in R^n. Of course you can identify a vector with a directional derivative, but this is not what one means generally by vector.
Now I see your point. In your view, a vector field X is basically the differential operator:
But you see, again, this identification of a vector field with the differential operator acting on the space of scalar functions and taking values in the space scalar functions, is not what people mean by a vector field (unless you are deep in differential geometry). And you did not explain this identification. It is no surprise that I did not understand the section you just added. Oleg Alexandrov 00:58, 5 Jun 2005 (UTC)
A scalar field is a tensor of rank (0,0). However, the function (x,y)-> x is not a tensor and thus not a scalar field. (It is not invariant under rotations, but a scalar field must be invariant, because a tensor of rank (0,0) is by definition invariant under rotations.) — MFH: Talk 16:32, 20 Jun 2005 (UTC)
This is a terrible way to explain "the" difference between vector fields and scalar fields. Okay, it is a diifference between them, but it is certainly not the essence of what makes them different; rather it is a consequence of that essential difference. The gist of their difference is that vectors and scalars have different meanings: a vector describes a direction in which a particle can move away from a location; a scalar is a number (or if you insist, a tuple of numbers) associated to a location. Daqu ( talk) 05:52, 10 May 2008 (UTC)
Okay, a full explanation is now in place. Perhaps you really like it. I feel it is rather unencyclopedic at the moment, but maybe that is only because of the popular way I phrased it and you disagree. -- MarSch 18:43, 11 Jun 2005 (UTC)
I moved some stuff contributed by MarSch in scalar field to here, as it has more to do with vectors than with scalars.
From what I know, a vector does not change if you change coordinates, only its components change. This unlike what the section ===Example 2=== in this article seems to imply.
Also, a vector is not a differential operator. In some situations it is convenient to identify a vector with the partial derivative in the direction of the vector, but this identification is by no means universal in mathematics. And without this identification (which was not made explicit), the section ===Example 2=== from this article does not make sence. Any comments? Oleg Alexandrov 04:45, 25 Jun 2005 (UTC)
Please correct me if I misunderstand something. So, the most elementary definition of a 2d vector is the arrow pointing from (0,0) to a point P. All of us know that, and there is no disagreement of this matter altogether. Oleg Alexandrov 21:44, 26 Jun 2005 (UTC)
So, what you are saying is that by default R^2 is just a manifold. You can't talk about vectors till you are given an origin, or otherwise a coordinate system. Did I understand you correctly? Oleg Alexandrov 21:44, 26 Jun 2005 (UTC)
OF COURSE a vector field is not the same as a bunch of scalar fields! If a manifold M has dimension n, then the tangent space to M is not isomorphic to the manifold
even if both of them have dimension 2n.
But OF COURSE that is not clear from the section ==Difference between vector fields and scalar fields== because MarSch never bothered to say that a vector field no longer goes from R^n to R^n, but rather, from R^n to its tangent space. Oleg Alexandrov 28 June 2005 15:56 (UTC)
streaklines, fieldlines and pathlines. They seem all the same to me. Can anyone explain their difference? -- MarSch 12:48, 30 August 2005 (UTC)
I think this should be included in the article, though I'm not knowledgeable enough to do so. http://mathworld.wolfram.com/Turbine.html Cako 01:36, 3 October 2007 (UTC)
Recently, a rather awful image was added to the lead of this article, that is totally irrelevant for the subject. I reverted this edit. Subsequently, my edit in turn was reverted by the same editor who added the image, with the edit summary:
Now, an editor who just added this article to three different articles is now edit warring to include it in two of them to ensure "consistency"? This seems to be very disingenuous reasoning. I vote to remove the image in question from the articles vector field and scalar field (I will leave it in tensor field). Are there any objections, besides the one already offered by the editor who is lobbying for inclusion of the image? I have posted to WT:WPM to garner further input. Sławomir Biały ( talk) 20:01, 13 June 2010 (UTC)
![]() | This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 |
![]() | This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 |
Shouldn't the vectors in the picture for vector field on a sphere be tangent vectors? —Preceding unsigned comment added by 134.226.81.3 ( talk) 14:45, 21 April 2009 (UTC)
The introduction is rather good as is. However, I think it would be important to draw an analogy, as the reader can have a picture in his mind before attacking the maths. I'm sure my 14 year-old self would have liked it.
If there isn't any disapproval, I'd like to quote the own article and expand the second sentence to:
Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point. They can be compared to scalar fields, which simply associate a number or scalar to every point in space, such as the temperature in a room.
Also, my bogus-o-meter bounces when reading "a vector field [...] associates a vector to every point in [...] Euclidean space." I can sure imagine a vector field in non-Euclidian space... Thoughts? FelisSchrödingeris ( talk) 00:58, 17 January 2008 (UTC)
There seems to be a mistake in the relation given in Section 1.1 for translating the expression of a vector field from one coordinate system to another. I think it should read:
.
This issue has been discussed in a Usenet thread. Could someone please verify this? Thanks. -- Pouya Tafti 20:56, 30 November 2006 (UTC)
This page needs an explanation of vector fields acting as operators on functions since this concept is used elsewhere e.g. Lie algebra - Gauge 04:02, 3 Aug 2004 (UTC)
If you think a page should be expanded on then be bold and do it yourself. There is no need to discuss it on the talk page. On the other hand if you are completely restructering a page a short discussion on the talk page might be apropriate to prevent an edit war. MathMartin 10:08, 3 Aug 2004 (UTC)
This stuff is at Tangent_vector#Tangent_vectors_as_directional_derivatives. The problem is that we have too many pages with unclear relations to eachother. -- MarSch 17:31, 11 Jun 2005 (UTC)
This article is part of a series of closely related articles for which I would like to clarify the interrelations. Please contribute your ideas at Wikipedia talk:WikiProject Mathematics/related articles. -- MarSch 14:05, 12 Jun 2005 (UTC)
The following paragraph is now in the article:
The difference between a scalar and a vector field is in how their coordinates respond to coordinate transformations. Coordinates of scalars, by definition, don't transform at all. See example for Euclidean and cylindrical coordinates:
I beg to contradict you. A vector is a quantity with n components, an element in R^n. Of course you can identify a vector with a directional derivative, but this is not what one means generally by vector.
Now I see your point. In your view, a vector field X is basically the differential operator:
But you see, again, this identification of a vector field with the differential operator acting on the space of scalar functions and taking values in the space scalar functions, is not what people mean by a vector field (unless you are deep in differential geometry). And you did not explain this identification. It is no surprise that I did not understand the section you just added. Oleg Alexandrov 00:58, 5 Jun 2005 (UTC)
A scalar field is a tensor of rank (0,0). However, the function (x,y)-> x is not a tensor and thus not a scalar field. (It is not invariant under rotations, but a scalar field must be invariant, because a tensor of rank (0,0) is by definition invariant under rotations.) — MFH: Talk 16:32, 20 Jun 2005 (UTC)
This is a terrible way to explain "the" difference between vector fields and scalar fields. Okay, it is a diifference between them, but it is certainly not the essence of what makes them different; rather it is a consequence of that essential difference. The gist of their difference is that vectors and scalars have different meanings: a vector describes a direction in which a particle can move away from a location; a scalar is a number (or if you insist, a tuple of numbers) associated to a location. Daqu ( talk) 05:52, 10 May 2008 (UTC)
Okay, a full explanation is now in place. Perhaps you really like it. I feel it is rather unencyclopedic at the moment, but maybe that is only because of the popular way I phrased it and you disagree. -- MarSch 18:43, 11 Jun 2005 (UTC)
I moved some stuff contributed by MarSch in scalar field to here, as it has more to do with vectors than with scalars.
From what I know, a vector does not change if you change coordinates, only its components change. This unlike what the section ===Example 2=== in this article seems to imply.
Also, a vector is not a differential operator. In some situations it is convenient to identify a vector with the partial derivative in the direction of the vector, but this identification is by no means universal in mathematics. And without this identification (which was not made explicit), the section ===Example 2=== from this article does not make sence. Any comments? Oleg Alexandrov 04:45, 25 Jun 2005 (UTC)
Please correct me if I misunderstand something. So, the most elementary definition of a 2d vector is the arrow pointing from (0,0) to a point P. All of us know that, and there is no disagreement of this matter altogether. Oleg Alexandrov 21:44, 26 Jun 2005 (UTC)
So, what you are saying is that by default R^2 is just a manifold. You can't talk about vectors till you are given an origin, or otherwise a coordinate system. Did I understand you correctly? Oleg Alexandrov 21:44, 26 Jun 2005 (UTC)
OF COURSE a vector field is not the same as a bunch of scalar fields! If a manifold M has dimension n, then the tangent space to M is not isomorphic to the manifold
even if both of them have dimension 2n.
But OF COURSE that is not clear from the section ==Difference between vector fields and scalar fields== because MarSch never bothered to say that a vector field no longer goes from R^n to R^n, but rather, from R^n to its tangent space. Oleg Alexandrov 28 June 2005 15:56 (UTC)
streaklines, fieldlines and pathlines. They seem all the same to me. Can anyone explain their difference? -- MarSch 12:48, 30 August 2005 (UTC)
I think this should be included in the article, though I'm not knowledgeable enough to do so. http://mathworld.wolfram.com/Turbine.html Cako 01:36, 3 October 2007 (UTC)
Recently, a rather awful image was added to the lead of this article, that is totally irrelevant for the subject. I reverted this edit. Subsequently, my edit in turn was reverted by the same editor who added the image, with the edit summary:
Now, an editor who just added this article to three different articles is now edit warring to include it in two of them to ensure "consistency"? This seems to be very disingenuous reasoning. I vote to remove the image in question from the articles vector field and scalar field (I will leave it in tensor field). Are there any objections, besides the one already offered by the editor who is lobbying for inclusion of the image? I have posted to WT:WPM to garner further input. Sławomir Biały ( talk) 20:01, 13 June 2010 (UTC)
![]() | This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 |