...side.

In skew Cartesian frames, i.e. non-orthogonal, --note that a frame is a set of basis vectors forming a coordinate system-- the projection values of a vector onto the basis vectors forming the skew frame are called the covariant coordinates of the vector. The covariant coordinates form the covariant vector representation of the original vector. In a skew frame, in contrast to an orthogonal frame, it is generally not possible to reconstruct the original vector from the covariant coordinates and the basis vectors. Reconstructing the original vector requires to know the set of contravariant coordinates of the vector, which can be calculated from the basis vectors and the covariant coordinates. The contravariant coordinates are the usual coordinates that we are used to manipulate in orthogonal Cartesian frames to construct vectors. In such frames, the covariant and contravariant coordinates are actually the same.

Olivier Coenen
Sun Aug 29 18:33:14 PDT 1999