Covariantit

Covariantit is a term used in mathematics and physics to describe the property of transforming covariantly under changes of coordinates. In this sense, it is closely related to the broader concept of covariance in tensor calculus, where objects are described in a way that respects the structure of the space they inhabit.

Covariant objects transform according to the covariant transformation law. Typically, components with lower indices, known as covariant tensors or covectors, transform with the Jacobian of the coordinate change. If x' = f(x) is a change of coordinates, a covariant object with indices μ, ν, ... transforms as T'_{μν...} = (∂x^α/∂x'^μ)(∂x^β/∂x'^ν)... T_{αβ...}. This is in contrast to contravariant objects, whose components carry upper indices and transform with the inverse Jacobian. Mixed tensors combine both behaviors.

Common examples help illustrate covariantit. The gradient of a scalar field, ∂_μ φ, is a covariant vector (a covector). The metric tensor g_{μν} is a rank-2 covariant tensor, providing a way to measure angles and lengths that remains well-behaved under coordinate changes. The differentials dx^μ form a basis of covariant components. The covariant derivative ∇ is designed to differentiate tensors while preserving their covariant transformation properties, ensuring the result remains a tensor.

In physics, the principle of general covariance requires that physical laws be expressed in a covariant form, independent of the chosen coordinate system. This is a cornerstone of general relativity, where field equations are written in terms of tensors to maintain form-invariance under diffeomorphisms.

Terminology varies: some use covariantit informally to denote covariance in a general sense, while the precise mathematical language refers to covariant tensors or covariant objects with lower indices.