Contravariante

Contravariante, or contravariant, is a term used in differential geometry and tensor analysis to describe components of tensors that transform with the inverse of a coordinate change. In a given coordinate system, a contravariante object typically carries upper indices and is associated with tangent vectors, while covariant objects carry lower indices and relate to cotangent vectors.

Under a coordinate transformation x -> x', with Jacobian matrix J^i_k = ∂x'^i/∂x^k, the contravariante components transform as V'^i = ∂x'^i/∂x^k V^k. In contrast, covariante (covariant) components transform with the forward Jacobian: A'_i = ∂x^k/∂x'^i A_k. This distinction reflects that contravariante components live in the tangent space, while covariant components live in the cotangent space.

A basic example is a vector field V, written in components as V = V^i ∂/∂x^i. The V^i are contravariante components. The dual object, a 1-form ω, has covariante components ω_i such that ω = ω_i dx^i. The metric g provides a link between these spaces: it can raise a contravariante index to a covariante one via V^i = g^{ij} V_j, and lower a covariante index to a contravariante one via V_i = g_{ij} V^j.

For higher-rank tensors, a tensor of type (p, q) has p contravariant indices and q covariant indices, transforming with p inverse Jacobians and q forward Jacobians accordingly.

Contravariante components are a central part of expressing geometric objects in a coordinate-independent way, enabling clear distinction between transformations of vectors and covectors.