Contravariant and Covariant Vectors

The reasoning behind the names contravariant and covariant is not particularly important, and a little convoluted, and goes like this. A choice of coordinate system is essentially a choice of the reference matter used to define coordinate axes, and should make no difference to the matter being described. Suppose we have a 3-vector space defined by axes described by basis vectors, , (not necessarily orthogonal or normal) and we now decide to describe it in terms of some other axes, represented by basis vectors . A given vector, , represents a property of some physical object, which is unchanged by a choice of different coordinates. The coefficients of must change in a way which opposes the change in the coordinates, so as to cancel it out. They are therefore called contravariant. On the other hand, the coefficients of a bra , also representing a property of a physical object, change in such a way as to oppose changes in contravariant coefficients, i.e. causes changes to cancel from the inner product. Since a double opposite cancels, the coefficients of a bra change in the same way as the coordinate axes, and are called covariant.

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