# Contravariant vs Covariant components - misprint?

1. Jan 16, 2012

### Evgn

In their article [Integrals in the theory of electron correlations, Annalen der Physik 7, 71] L.Onsager at el. write:

By resolving the vector $\vec{s}$ into its contravariant components in the oblique coordinate system formed by the vectors $\vec{q}$ and $\vec{Q}$ it is possible to reduce the region of integration (3.11) to a rectangle. The contravariant components of $\vec{s}$ are defined by:

$s_1 = \frac{\vec{s}\vec{q}}{|q|}, \quad s_2 = \frac{\vec{s}\vec{Q}}{|Q|}$

Shouldn't they write "covariant components"?

PDF of the article can be found here: http://zs.thulb.uni-jena.de/receive/jportal_jparticle_00133463

Thank you.

2. Jan 19, 2012

### Fredrik

Staff Emeritus
Why should it be "covariant"? I don't see a reason to mention either of those words. $s_1$ and $s_2$ are just the components of $\vec s$ in the ordered basis $\Big\langle\frac{\vec q}{|\vec q|},\frac{\vec Q}{|\vec Q|}\Big\rangle$.

However, if I had to choose one of those words, I'd go with "contravariant", because when $\{e_i\}$ is a basis for the tangent space of a manifold, a tangent vector v can be expressed as $v=v^i e_i$, and the $v^i$ are said to define a contravariant vector. (I strongly dislike this terminology, and don't understand why people are still using it). The "covariant components of $\vec v$" in this (horrible) terminology would be components of the 1-form $g(\vec v,\cdot)$ in the dual basis for the cotangent space that's the dual basis of $\{e_i\}$.

Also note that if the manifold is Euclidean, the contravariant and covariant components are the same.