What Is the Difference Between Covariant and Contravariant Vectors?

[codeman_nz]

Hi everyone,

I am having a little trouble with the difference between a covariant vector and contravariant vector. The examples that I come across say that an example of a contravariant vector is velocity and that a contravariant must contra-vary with a change of basis to compensate.

So if you have a velocity that is measured in m/s and you want to measure it in km/s then the velocity shrinks. Is that because you increased the co-ordinate system to km and so that velocity must shrink to compensate for that increase? Or am I thinking of it the wrong way?

 

[Matterwave]

More modern terminology has contravariant vectors simply being called vectors and covariant vectors being called one forms. Covariant vectors (one forms) are best thought of, in my opinion, as simply functions of contravariant (vectors) vectors into real numbers. In other words, one forms are functions, e.g. f(), which accept a vector argument and returns a number: f(v)=real number.

The two are dual to each other, and are defined in the classical literature by the transformation properties of their components under coordinate transformations. More recently, one tends to work more conceptually and equate vectors with either an equivalence classes of curves (1-1 mappings from R into the manifold) or with differential operators because there is a 1-1 isomorphism between these concepts. One forms are then simply defined and thought of as above, i.e., functions of vectors into real numbers.