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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, in changing scale from meters to centimeters, the components of a measured velocity vector will multiply by 100. Vectors exhibit this behavior of changing scale inversely to changes in scale to the reference axes: they are contravariant. As a result, vectors often have units of distance or distance times some other unit. In contrast, dual vectors typically have units the inverse of distance or the inverse of distance times some other unit. An example of a dual vector is the gradient, which has units of a spatial derivative, or distance−1. The components of dual vectors change in the same way as changes to scale of the reference axes: they are covariant. The components of vectors and covectors also transform in the same manner under more general changes in basis: Wikipedia

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