So I'm just a computer scientist, not a mathematician or phycist, but I don't think there is a direct connection. The words co/contravariance is used for two di

So I'm just a computer scientist, not a mathematician or phycist, but I don't think there is a direct connection. The words co/contravariance is used for two different things in mathematics, vectors in physics/differential geometry, and functors in category theory. (https://en.wikipedia.org/wiki/Covariance_and_contravariance https://en.wikipedia.org/wiki/Covariance_and_contravariance). The computer science meaning is inspired by category theory (and can be seen as a simple instance of it). But the vector/functor meanings seem to be unrelated. There is a bit of a connection, because the operation ٭ which takes a vector space V to the set of 1-forms V٭ is a contravariant functor. (I.e., linear maps V→U give maps of 1-forms U٭→V٭). But in physics, we say that the members of V are contravariant and the members of V٭ are covariant, i.e. the physics notion is about individual spaces, not about mappings between spaces.