- #1

Trying2Learn

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- TL;DR Summary
- Is there an error in wikipedia

Good Morning!

I understand that a vector is a physical object

I understand that it is the underlying basis that determines how the components transform.

However, I encounter this:

https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors

The fifth paragraph has this statement

A contravariant vector or tangent vector (often abbreviated simply as

I am confused.

This seems to be suggesting that of the various types of vectors, one vector, the

This makes no sense and is contrary to my understanding that a vector (or, say, even the stress tensor) does NOT have ONE manifestation (the latter could be contra, co or mixed)

Am I reading too much into the wiki statement?

Because I can use the metric tensor and convert a contravariant position vector to be one with its dual basis and covariant components.

I understand that a vector is a physical object

I understand that it is the underlying basis that determines how the components transform.

However, I encounter this:

https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors

The fifth paragraph has this statement

A contravariant vector or tangent vector (often abbreviated simply as

*vector*, such as a direction vector or velocity vector) has components that*contra-vary*with a change of basis to compensate. That is, the matrix that transforms the vector components must be the inverse of the matrix that transforms the basis vectors. The components of vectors (as opposed to those of covectors) are said to be**contravariant**.I am confused.

This seems to be suggesting that of the various types of vectors, one vector, the

**direction vector**(and, I assume velocity, acceleration) KNOWS itself to be a contravariant vector.This makes no sense and is contrary to my understanding that a vector (or, say, even the stress tensor) does NOT have ONE manifestation (the latter could be contra, co or mixed)

Am I reading too much into the wiki statement?

Because I can use the metric tensor and convert a contravariant position vector to be one with its dual basis and covariant components.