Covariance and Contravariance of Vectors

Covariance And Contravariance Of Vectors

• tangent basis vectors (yellow, left: e₁, e₂, e₃) to the coordinate curves (black),

• dual basis, covector basis, or cobasis (blue, right: e1, e2, e3), normal vectors to coordinate surfaces (grey),

In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. For holonomic bases, this is determined by a change from one coordinate system to another. When an orthogonal basis is rotated into another orthogonal basis, the distinction between co- and contravariance is invisible. However, when considering more general coordinate systems such as skew coordinates, curvilinear coordinates, and coordinate systems on differentiable manifolds, the distinction is significant.

• For a vector (such as a direction vector or velocity vector) to be basis-independent, the components of the vector must contra-vary with a change of basis to compensate. That is, the components must vary with the inverse transformation to that of the change of basis. The components of vectors (as opposed to of dual vectors) are said to be contravariant. Examples of vectors with contravariant components include the position of an object relative to an observer, or any derivative of position with respect to time, including velocity, acceleration, and jerk. In Einstein notation, contravariant components are denoted with upper indices as in

• For a dual vector (also called a covector) to be basis-independent, the components of the dual vector must co-vary with a change of basis to remain representing the same covector. That is, the components must vary by the same transformation as the change of basis. The components of dual vectors (as opposed to of vectors) are said to be covariant. Examples of covariant vectors generally appear when taking a gradient of a function. In Einstein notation, covariant components are denoted with lower indices as in

In physics, vectors often have units of distance or distance times some other unit (such as the velocity), whereas covectors have units the inverse of distance or the inverse of distance times some other unit. The distinction between covariant and contravariant vectors is particularly important for computations with tensors, which can have mixed variance. This means that they have components that are both covariant and contravariant. The valence or type of a tensor gives the number of covariant and contravariant component indices.

The terms covariant and contravariant were introduced by J.J. Sylvester in 1853 in order to study algebraic invariant theory. In this context, for instance, a system of simultaneous equations is contravariant in the variables. The use of both terms in the modern context of multilinear algebra is a specific example of corresponding notions in category theory.