Covariant Transformation

Covariant Transformation

The co/contra-variant nature of vector coordinates has been treated as an elementary characterisation in Tensor Analysis. For example in the classic text translated from the Russian 1966 3rd Edition and published by Dover, the coordinate space is first introduced with coordinates in superscript indices (xi) and oblique reference frame and basis vectors with subscript indices ; and the motivation for this convention is deferred to following text quoted later:

"These designations of the components of a vector stem from the fact that the direct transformation of the covariant components involves the coefficients αk_i' of the direct transformation, that is A'_i= αk_i'A_k. while the direct transformation of the contravariant components involve the coefficients αi'_k of the inverse transformation A'i = αi'_kAk."

In the first instance, suppose f is a function over vector space, one can express the scalar derivative components of f in new coordinates in terms of the old coordinates using the chain rule and get

Direct differentiation of the coordinate values produces a transformation with each k-i element, where the transformed (new) bases equal the rate of change of the old (x) bases with respect to the new (x') coordinates, times the old bases. To paraphrase, transforms as change of old bases times the old, (transform directly, component index in subscript).

In the second case where the components are not coordinates but some derivative of the coordinate such that vi = dxi/dλ, when we perform a change of bases, for each new coordinate component (i), xi, it fixes relative to independent scalar components (j), by the chain rule

namely, that the new bases equal the rate of change of the new (x') coordinates with respect to the old (x) coordinates, times the old bases. To paraphrase, transforms as change of new bases times the old (transform inversely, component index in superscript).