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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.
In physics, a basis is sometimes thought of as a set of reference axes. A change of scale on the reference axes corresponds to a change of units in the problem. For instance, by changing scale from meters to centimeters , the components of a measured velocity vector are multiplied by 100. A vector changes scale inversely to changes in scale to the reference axes, and consequently is called contravariant. As a result, a vector often has units of distance or distance with other units.
In contrast, a covector, also called a dual vector, typically has units of the inverse of distance or the inverse of distance with other units. For example, a gradient which has units of a spatial derivative, or distance. The components of a covector changes in the same way as changes to scale of the reference axes, and consequently is called covariant.
A third concept related to covariance and contravariance is invariance. An example of a physical observable that does not change with a change of scale on the reference axes is the mass of a particle, which has units of mass. The single, scalar value of mass is independent of changes to the scale of the reference axes and consequently is called invariant.