Back Covariància i contravariància de vectors Catalan Συνδιακύμανση και αντιδιακύμανση διανυσμάτων Greek Covarianza y contravarianza Spanish Vecteur contravariant, covariant et covecteur French וקטורים קוואריאנטיים וקונטרוואריאנטיים HE Կովարիանտ և կոնտրավարիանտ թենզոր Armenian Covarianza e controvarianza Italian ベクトルの共変性と反変性 Japanese 벡터의 공변성 및 반변성 Korean Contravariant en covariant Dutch

Covariance and contravariance of vectors

For use of "covariance" in the context of special relativity, see Lorentz covariance. For other uses of "covariant" or "contravariant", see Covariance and contravariance (disambiguation).
A   vector, v, represented in terms of
tangent basis
  e1, e2, e3 to the   coordinate curves (left),
dual basis, covector basis, or reciprocal basis
  e1, e2, e3 to   coordinate surfaces (right),
in 3-d general curvilinear coordinates (q1, q2, q3), a tuple of numbers to define a point in a position space. Note the basis and cobasis coincide only when the basis is orthogonal.[1]

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. [2] Briefly, a contravariant vector is a list of numbers that transforms oppositely to a change of basis, and a covariant vector is a list of numbers that transforms in the same way. Contravariant vectors are often just called vectors and covariant vectors are called covectors or dual vectors. The terms covariant and contravariant were introduced by James Joseph Sylvester in 1851.[3][4]

Curvilinear coordinate systems, such as cylindrical or spherical coordinates, are often used in physical and geometric problems. Associated with any coordinate system is a natural choice of coordinate basis for vectors based at each point of the space, and covariance and contravariance are particularly important for understanding how the coordinate description of a vector changes by passing from one coordinate system to another. Tensors are objects in multilinear algebra that can have aspects of both covariance and contravariance.

  1. ^ Misner, C.; Thorne, K.S.; Wheeler, J.A. (1973). Gravitation. W.H. Freeman. ISBN 0-7167-0344-0.
  2. ^ Frankel, Theodore (2012). The geometry of physics : an introduction. Cambridge: Cambridge University Press. p. 42. ISBN 978-1-107-60260-1. OCLC 739094283.
  3. ^ Sylvester, J.J. (1851). "On the general theory of associated algebraical forms". Cambridge and Dublin Mathematical Journal. Vol. 6. pp. 289–293.
  4. ^ Sylvester, J.J. University Press. The collected mathematical papers of James Joseph Sylvester. Vol. 3, 1870–1883. ISBN 978-1107661431. OCLC 758983870.

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