covariant basis

We'll favor writing quantities with respect to the covariant basis.

These represent the covariant basis; computing their dot products gives the following components of the metric tensor:

As before, are covariant basis vectors and b, b are contravariant basis vectors.

The basis vectors shown above are covariant basis vectors (because they "co-vary" with vectors).

Likewise, the covariant components of v can be obtained from the dot product of v with covariant basis vectors, viz.

We now face three different basis sets commonly used to describe vectors in orthogonal coordinates: the covariant basis e, the contravariant basis e, and the normalized basis ê.

If the contravariant basis vectors are orthonormal then they are equivalent to the covariant basis vectors, so there is no need to distinguish between the covariant and contravariant coordinates.

From the above vector sums, it can be seen that contravariant coordinates are associated with covariant basis vectors, and covariant coordinates are associated with contravariant basis vectors.