Covariance and Contravariance of Vectors - Coordinates

Coordinates

The choice of basis f on the vector space V defines uniquely a set of coordinate functions on V, by means of

The coordinates on V are therefore contravariant in the sense that

Conversely, a system of n quantities vi that transform like the coordinates xi on V defines a contravariant vector. A system of n quantities that transform oppositely to the coordinates is then a covariant vector.

This formulation of contravariance and covariance is often more natural in applications in which there is a coordinate space (a manifold) on which vectors live as tangent vectors or cotangent vectors. Given a local coordinate system xi on the manifold, the reference axes for the coordinate system are the vector fields

This gives rise to the frame f = (X₁,...,X_n) at every point of the coordinate patch.

If yi is a different coordinate system and

then the frame f' is related to the frame f by the inverse of the Jacobian matrix of the coordinate transition:

Or, in indices,

A tangent vector is by definition a vector that is a linear combination of the coordinate partials . Thus a tangent vector is defined by

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

Therefore the components of a tangent vector transform via

Accordingly, a system of n quantities vi depending on the coordinates that transform in this way on passing from one coordinate system to another is called a contravariant vector.