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- Covariance_and_contravariance_of_vectors abstract "For other uses of "covariant" or "contravariant", see covariance and contravariance (disambiguation).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 (that is, dividing the scale of the reference axes by 100), 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 (like the velocity). In contrast, dual vectors (also called covectors) 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. Vectors and covectors also transform in the same manner under more general changes in basis: For a vector (such as a direction vector or velocity vector) to be basis-independent, the components of the vector must contra-vary with a change of basis to compensate. That is, the transformation that acts on the vector of components must be the inverse of the transformation that acts on the basis vectors. The components of vectors (as opposed to those of dual vectors) are said to be contravariant. Examples of vectors with contravariant components include the position of an object relative to an observer, or any derivative of position with respect to time, including velocity, acceleration, and jerk. In Einstein notation, contravariant components are denoted with upper indices as in For a dual vector (also called a covector) to be basis-independent, the components of the dual vector must co-vary with a change of basis to remain representing the same covector. That is, the components must vary by the same transformation as the change of basis. The components of dual vectors (as opposed to those of vectors) are said to be covariant. Examples of covariant vectors generally appear when taking a gradient of a function. In Einstein notation, covariant components are denoted with lower indices as inCurvilinear coordinate systems, such as cylindrical or spherical coordinates, are often used in physical and geometric problems. Associated to 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 in passing from one coordinate system to another.The terms covariant and contravariant were introduced by J.J. Sylvester in 1853 in order to study algebraic invariant theory. In this context, for instance, a system of simultaneous equations is contravariant in the variables. Tensors are objects in multilinear algebra that can have aspects of both covariance and contravariance. The use of both terms in the modern context of multilinear algebra is a specific example of corresponding notions in category theory.".
- Covariance_and_contravariance_of_vectors thumbnail Vector_1-form.svg?width=300.
- Covariance_and_contravariance_of_vectors wikiPageExternalLink kmath398.htm.
- Covariance_and_contravariance_of_vectors wikiPageID "202886".
- Covariance_and_contravariance_of_vectors wikiPageRevisionID "606767536".
- Covariance_and_contravariance_of_vectors hasPhotoCollection Covariance_and_contravariance_of_vectors.
- Covariance_and_contravariance_of_vectors id "p/c025970".
- Covariance_and_contravariance_of_vectors id "p/c026880".
- Covariance_and_contravariance_of_vectors title "Contravariant Tensor".
- Covariance_and_contravariance_of_vectors title "Contravariant tensor".
- Covariance_and_contravariance_of_vectors title "Covariant Tensor".
- Covariance_and_contravariance_of_vectors title "Covariant tensor".
- Covariance_and_contravariance_of_vectors urlname "ContravariantTensor".
- Covariance_and_contravariance_of_vectors urlname "CovariantTensor".
- Covariance_and_contravariance_of_vectors subject Category:Differential_geometry.
- Covariance_and_contravariance_of_vectors subject Category:Riemannian_geometry.
- Covariance_and_contravariance_of_vectors subject Category:Tensors.
- Covariance_and_contravariance_of_vectors subject Category:Vectors.
- Covariance_and_contravariance_of_vectors comment "For other uses of "covariant" or "contravariant", see covariance and contravariance (disambiguation).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.".
- Covariance_and_contravariance_of_vectors label "Covariance and contravariance of vectors".
- Covariance_and_contravariance_of_vectors label "Covariancia y contravariancia".
- Covariance_and_contravariance_of_vectors label "Covarianza e controvarianza".
- Covariance_and_contravariance_of_vectors label "Контравариантный вектор".
- Covariance_and_contravariance_of_vectors label "共變和反變".
- Covariance_and_contravariance_of_vectors sameAs Covariancia_y_contravariancia.
- Covariance_and_contravariance_of_vectors sameAs Covarianza_e_controvarianza.
- Covariance_and_contravariance_of_vectors sameAs ベクトルの共変性と反変性.
- Covariance_and_contravariance_of_vectors sameAs m.01ctpf.
- Covariance_and_contravariance_of_vectors sameAs Q7253388.
- Covariance_and_contravariance_of_vectors sameAs Q7253388.
- Covariance_and_contravariance_of_vectors wasDerivedFrom Covariance_and_contravariance_of_vectors?oldid=606767536.
- Covariance_and_contravariance_of_vectors depiction Vector_1-form.svg.
- Covariance_and_contravariance_of_vectors isPrimaryTopicOf Covariance_and_contravariance_of_vectors.
