Covariant and Contravariant Vectors

TL;DR

This paper explains the concepts of covariant and contravariant vectors, illustrating their importance in physics and geometry, especially when working with non-orthogonal bases and non-Euclidean spaces.

Contribution

It clarifies the role of covariant and contravariant components in various contexts, including physics and differential geometry, highlighting their utility in handling non-orthogonal bases.

Findings

Covariant and contravariant vectors simplify calculations in non-orthogonal bases.

The concepts relate to quantum mechanics' Ket and Bra vectors.

Application to tangent vectors in non-Euclidean geometry.

Abstract

Vector is a physical quantity and it does not depend on any co-ordinate system. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. The expansion in orthonormal basis is mathematically simple. But in many physical situations we have to choose an non-orthogonal basis (or oblique co-ordinate system). But the expansion of a vector in non-orthogonal basis is not convenient to work with. With the notion of contravariant and covariant components of a vector, we make non-orthogonal basis to behave like orthonormal basis. The same notion appears in quantum mechanics as Ket and Bra vectors and we compare the two equivalent situation via the completeness relation. This notion appears in the differential geometry of a metric manifold for tangent vectors at a point, where it takes into account the non-orthogonality of basis as well as non-Euclidean geometry.