Granzer92

Help me!

On the page covering the topic on Covariance and contravariance of vectors, https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors, it has been said contravariance vectors have components that "transform as the coordinates do" and inversely to the transformation of the reference axes. What are the difference between (or the definition of) components of the vectors, coordinates of the vectors and reference axes of the vectors? Please help me with...

granzer

Questions unrelated to Wikipedia editing belong at the relevant reference desk; in this case, the question should be taken to Wikipedia:Reference desk/Mathematics. 78.28.45.127 (talk) 12:25, 9 June 2018 (UTC)

I thought you might like to know that I too had similar difficulty understanding the verbiage in treatises of contravariance, covariance, and tensors. After much research and frustration I finally determined that most linear algebra textbooks are not clear about the distinction between coordinates and components, and most tensor textbooks are not clear about the distinction between coordinates and basis vectors. Following your post (Confusion between vector components, vector coordinate and the reference axes) to Talk:Covariance and contravariance of vectors on 10 June 2018, I made a post (Standard definition apparent inconsistency) and eventually corrected it. Please take a look, as it may help to illuminate the answers to your questions. You may also want to see Euclidean vector#Decomposition or resolution.

As I understand, when the basis vectors (sometimes called "reference axes") are transformed, the coordinates (of everything else) change inversely. So a position vector transforms inversely as the basis vectors (its components are thus contravariant), and the same as the coordinates (which may seem like a tautology). A gradient vector transforms the same as the basis vectors (its components are thus covariant), and inversely as the coordinates.

The best definition of coordinates and components I have seen is the following:

If $\alpha =\sum _{i=1}^{n}a_{i}\alpha _{i}$, the scalar $a_{i}$ is called the i-th coordinate of $\alpha$, and $a_{i}\alpha _{i}$ is called the i-th component of $\alpha$. Generally, coordinates and components depend on the choice of the entire basis and cannot be determined from individual vectors in the basis. Because of the rather simple correspondence between coordinates and components there is a tendency to confuse them and to use both terms for both concepts. Since the intended meaning is usually clear from context, this is seldom a source of difficulty.

I hope this makes things more clear, rather than muddying the waters further.—Anita5192 (talk) 18:41, 28 November 2018 (UTC)

References