Covariance and contravariance of vectors

In physics, a vector typically arises as the outcome of a measurement or series of measurements, and is represented as a list (or tuple) of numbers such as

Covariant and contravariant components of a vector when the basis is not orthogonal.

However, since the vector v itself is invariant under the choice of basis,

The way A relates the two pairs is depicted in the following informal diagram using an arrow. The reversal of the arrow indicates a contravariant change:

The way A relates the two pairs is depicted in the following informal diagram using an arrow. A covariant relationship is indicated since the arrows travel in the same direction:

Had a column vector representation been used instead, the transformation law would be the transpose

Such a vector is contravariant with respect to change of frame. Under changes in the coordinate system, one has

Thus the change of basis matrix in going from the original basis to the reciprocal basis is

The covariant components are obtained by equating the two expressions for the vector v:

If the basis vectors are orthonormal, then they are the same as the dual basis vectors.

Despite this usage of "covariant", it is more accurate to say that the Klein–Gordon and Dirac equations are invariant, and that the Schrödinger equation is not invariant. Additionally, to remove ambiguity, the transformation by which the invariance is evaluated should be indicated.

Because the components of vectors are contravariant and those of covectors are covariant, the vectors themselves are often referred to as being contravariant and the covectors as covariant.