Algebraic Characterisation
The co/contra-variance of a transformation is an algebraic property, and the designation cited above also applies in a generalised differential geometry setting where given a category C and objects V, W belonging to C, a covariant functor L maps the set of morphisms Hom(V,W) to Hom(LV,LW), whereas a contravariant functor L' maps Hom(V,W) to Hom(L'W,L'V). Again notice the transformation domain's push forward direction in the covariant, and pull-back direction in the contravariant case. This construct extends the covariant differential to manifolds such as vector bundles and their connections, for example by parallel transport extending covariant derivatives to vector fields over manifolds by affinely connecting tangent vectors from one tangent bundle on the manifold to neighbouring fibres along a 'curve' (see for example covariant derivative).
The diagram following illustrates one such curve across three manifolds, from around the cone in conical chart X, then around the cylinder through cylindrical chart X' and finally along the rectilinear plane through rectangular chart X". The tangent vector representing a force along its transmission path, that is along the curve, is preserved by parallel transport, barring friction loss or geometric distortion of medium.
Read more about this topic: Covariant Transformation
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