| __
PlanetPhysics: covariance and contravariance__ *(Site not responding. Last check: 2007-10-26)* |

| | In the example of cylindrical coordinates, the radial and z components are the same in **covariant** and contravariant form, but the **covariant** component of the differential of angle round the z axis is r2dÎ¸ and its integral depends on the path. |

| | Homology theory is **covariant** because (as is very clear in singular homology) its basic construction is to take a topological space X and map things into it (in that case, simplices). |

| | By considering a coordinate transformation on a manifold as a map from the manifold to itself, the transformation of **covariant** indices of a tensor are given by a pullback, and the transformation properties of the contravariant indices is given by a pushforward. |

| planetphysics.org /encyclopedia/CovarianceAndContravariance.html (1326 words) |