Talk:Covariant transformation

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Contents 1 What is a good category for this article? 2 Merge 3 Coordinate Representation 4 Transformation versus change of coordinates

[edit] What is a good category for this article?

Oleg Alexandrov 22:20, 5 Mar 2005 (UTC)

Andrew Critch says:

This article gives a better explanation of covariance/contravariance than anything else I've seen on the net, including other Wikipedia articles. If any merging occurs, I think other articles involved should be (more or less) appended to the end of this one. This article article's introduction and examples should be left intact, and for the most part, at the BEGINNING of the merged article.

[edit] Merge

At first glance this article is better. —Ben FrantzDale 23:51, 30 August 2006 (UTC)

There are at least three related articles that are closely related and need cleanup: this one, covariance and contravariance, and differential form. Thoughts? —Ben FrantzDale 10:12, 11 May 2007 (UTC)

Leave differential form out of it. It is another example of pointless abstraction, like category theory. JRSpriggs 11:06, 11 May 2007 (UTC)

I also support merging of this one and covariance and contravariance. Both articles have their merits and demerits and a merged article will contribute toward completeness and clarity which is lacking in many points. Lantonov 14:32, 23 July 2007 (UTC)

[edit] Coordinate Representation

There needs to be a drawing of a set of arbitrary bases (duals), in the context of an orthogonal basis, representing the contravariant components alongside the covariant ones. I'll see if I can work on it.--Charlesrkiss 07:07, 28 October 2006 (UTC)

I plan to put soon (in maybe another week) such drawing in the article Curvilinear coordinates which I work on now. Lantonov 14:52, 23 July 2007 (UTC)

[edit] Transformation versus change of coordinates

I am a bit confused about covariance and contravariance. It looks like a covariant "transformation" is a change of coordinates and does not transform vectors at all. In the example (right), the vector v is represented in red in polar coordinates and in black in Cartesian coordinates, but both represent the same vector, v where v is in the tangent space of the point (3,4) (in Cartesian coordinates) (which is also the point (5, atan(4/3)) in polar coordinates). While the "transformation" from Cartesian to polar coordinates would change the numerical representation of v from (3/2,3/4) to something like (1.34, 1.067), we are still talking about the same vector.

But the text of the article says "The covariant transformation [from black to red] here is a clockwise rotation." But nothing is being rotated, right? —Ben FrantzDale 23:04, 28 April 2007 (UTC)

I can't see any counterclockwise rotation of the components, which are colinear to the basis vectors.(anonymous) —Preceding unsigned comment added by 90.7.103.35 (talk • contribs)

Exactly. —Ben FrantzDale 17:36, 17 May 2007 (UTC)

I think that the concept of rotations given here in an attempt to give a geometric interpretation of covariant and contravariant, is confusing. Personally, I couldn't see what is rotated in which direction, as the description is very ambiguous. It would be better to stick to the textbook description of co- and contravariant until someone finds more clear geometric interpretation. Otherwise, I am very ardent adherent to geometric clarification of obtuse math topics (covariance is one of them) and applaud any attempt in this direction, including this one. Lantonov 14:47, 23 July 2007 (UTC)

There is a very good geometrical treatment of this in a little book called "Geometirical Vectors" by Gabriel Weinrich.I found it very usefull and now think in terms of "arrow fields" and "stack fields" for contravariant and covarient "vectors".This aproach really is very helpful (at least I found it so)Dave59 09:57, 30 July 2007 (UTC)

It would be very helpful if you include this treatment from Weinrich book in the article. I am not aware of it. Lantonov 10:05, 30 July 2007 (UTC)

In the book (which is about 100 pages long) Weinreich uses geometrical arguments to formulate a version of traditional 3 dimensional vector calculus that is as far as possible topologically invariant. To do this he distinguishes between different types of “vector” that transform in different ways not only under rotations but also under “stretchings” and “compressions” of space. This leads to 4 different types of “vector” (covariant, contravariant, contravariant density and covariant capacity). It also becomes necessary to distinguish between scalar densities and scalar capacities. If this is done and then 1 co-ordinate system is defined metrically the algebraic forms of all the usual vector calculus identities become identical in all co ordinate systems as long as each type of vector and scalar is expressed in terms of its natural basis in terms of partial differentials as explained in this article i.e. contravariant vectors are given a co variant basis and vice versa. This is complicated and requires a lot of 3 d diagrams. I am not up to expressing it succinctly. Dave59 11:26, 30 July 2007 (UTC)

I am familiar with "contravariant density" and "covariant capacity" but not with their geometric interpretations. I can imagine that the whole thing is complicated, especially when it involves developing new constructs. Lantonov 11:38, 30 July 2007 (UTC)