Contravariant – Knowledge and References

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Tensors In General Coordinates

Published in C. Young Eutiquio, Vector and Tensor Analysis, 2017

Actually, the terms covariant and contravariant have to do with the manner in which the components of the vector transform under a change of oblique coordinate systems. To see how the components transform, suppose x*i is another oblique coordinate system determined by a basis e*i (1 ≤ i ≤ 3). Let a*i and a*i denote the components of the vector A with respect to the bases e*i and e*i, respectively. Then we have

Formulation of 3D finite elements using curvilinear coordinates

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Journal Information

Published in Mechanics of Advanced Materials and Structures, 2022

Maria Cinefra

Let us consider next a vector field in space and denote the vector by We define components of the vector u by resolving it in the directions of at the point P as follows: where is called the contravariant component of the vector u. Components of the vector u may be expressed alternatively by resolving it in the directions of at the point P: where is called the covariant component of the vector u and is given from Eqs. (7), (11) and (12) by: