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- Covariant_transformation abstract "See also Covariance and contravariance of vectorsIn physics, a covariant transformation is a rule (specified below) that specifies how certain entities change under a change of basis. In particular, the term is used for vectors and tensors. The transformation that describes the new basis vectors as a linear combination of the old basis vectors is defined as a covariant transformation. Conventionally, indices identifying the basis vectors are placed as lower indices and so are all entities that transform in the same way. The inverse of a covariant transformation is a contravariant transformation. Since a vector should be invariant under a change of basis, its components must transform according to the contravariant rule. Conventionally, indices identifying the components of a vector are placed as upper indices and so are all indices of entities that transform in the same way. The sum over pairwise matching indices of a product with the same lower and upper indices are invariant under a transformation.A vector itself is a geometrical quantity, in principle, independent (invariant) of the chosen basis. A vector v is given, say, in components vi on a chosen basis ei. On another basis, say e′j, the same vector v has different components v′i andWith v being invariant and the ei transforming covariantly, it must be that the vi (the set of numbers identifying the components) transform in a different way, being the inverse called the contravariant transformation rule.If, for example in a 2-dimensional Euclidean space, the new basis vectors are rotated anti-clockwise with respect to the old basis vectors, then it will appear in terms of the new system that the componentwise representation of the vector look as if the vector was rotated in the opposite direction, i.e. clockwise (see figure).A vector v is described in a given coordinate grid (black lines) on a basis which are the tangent vectors to the (here rectangular) coordinate grid. The basis vectors are ex and ey. In another coordinate system (dashed and red), the new basis vectors are tangent vectors in the radial direction and perpendicular to it. These basis vectors are indicated in red as er and eφ. They appear rotated anticlockwise with respect to the first basis. The covariant transformation here is thus an anticlockwise rotation. If we view the vector v with eφ pointed upwards, its representation in this frame appears rotated to the right. The contravariant transformation is a clockwise rotation. .".
- Covariant_transformation wikiPageID "443235".
- Covariant_transformation wikiPageRevisionID "594958730".
- Covariant_transformation hasPhotoCollection Covariant_transformation.
- Covariant_transformation subject Category:Differential_geometry.
- Covariant_transformation subject Category:Tensors.
- Covariant_transformation type Abstraction100002137.
- Covariant_transformation type Cognition100023271.
- Covariant_transformation type Concept105835747.
- Covariant_transformation type Content105809192.
- Covariant_transformation type Idea105833840.
- Covariant_transformation type PsychologicalFeature100023100.
- Covariant_transformation type Quantity105855125.
- Covariant_transformation type Tensor105864481.
- Covariant_transformation type Tensors.
- Covariant_transformation type Variable105857459.
- Covariant_transformation comment "See also Covariance and contravariance of vectorsIn physics, a covariant transformation is a rule (specified below) that specifies how certain entities change under a change of basis. In particular, the term is used for vectors and tensors. The transformation that describes the new basis vectors as a linear combination of the old basis vectors is defined as a covariant transformation.".
- Covariant_transformation label "Covariant et contravariant".
- Covariant_transformation label "Covariant transformation".
- Covariant_transformation label "Covariante".
- Covariant_transformation label "Kovarianz (Physik)".
- Covariant_transformation sameAs Kovarianz_(Physik).
- Covariant_transformation sameAs Συναλλοίωτος_μετασχηματισμός.
- Covariant_transformation sameAs Covariant_et_contravariant.
- Covariant_transformation sameAs Covariante.
- Covariant_transformation sameAs m.0293f2.
- Covariant_transformation sameAs Q362640.
- Covariant_transformation sameAs Q362640.
- Covariant_transformation sameAs Covariant_transformation.
- Covariant_transformation wasDerivedFrom Covariant_transformation?oldid=594958730.
- Covariant_transformation isPrimaryTopicOf Covariant_transformation.