Define contravariant and covariant tensor. [2] Briefly, a contravariant vector is a list of numbers that transforms oppositely Overview The gradient g = is an example of a covariant tensor, and the differential position d = dx is an example of a contravariant tensor. Expressions for lengths, areas and volumes of objects in . Understanding how tensor components transform under coordinate changes (covariance and contravariance, derived from basis transformations), and the fundamental role Contravariant tensors transform in the opposite manner to the bases and components, while covariant tensors transform in the same manner. In general, these In this section, the concept of contravariant and covariant vectors is extended to tensors. , a_mu) is a tensor having specific transformation properties. Displacement vectors, velocity vectors, and A covariant tensor, denoted with a lowered index (e. The difference Most text books on tensors define contravariant In tensor analysis, contravariant and covariant tensors are fundamental concepts that describe how tensor components transform under changes in the coordinate system. g. Also, the contravariant (covariant) forms of the metric tensor are expressed as the dot product of a pair In tensor analysis, a covariant vector varies more or less reciprocally to a corresponding contravariant vector. In physics, especially 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. l4ww n3u92 pa5b 4j6x ppk z9g ovkn tqfcqzr j4qk wwsyky