Covariance and Contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. They are the mathematical foundation to understand General Relativity.
【James Joseph Sylvester(1814-1897)】
The terms Covariant and Contravariant were introduced by James Joseph Sylvester in 1853 to study algebraic invariant theory. Now they are extensive using in Tensor Field analysis of mathematics, physics, and engineering, even in General Relativity. In order to understand what Covariance and Contravariance are in a vector space, in the beginning, a vector can be written as