Two-vector (original) (raw)
A two-vector or bivector is a tensor of type and it is the dual of a two-form, meaning that it is a linear functional which maps two-forms to the real numbers (or more generally, to scalars). The tensor product of a pair of vectors is a two-vector. Then, any two-form can be expressed as a linear combination of tensor products of pairs of vectors, especially a linear combination of tensor products of pairs of basis vectors. If f is a two-vector, then , An example of a bivector is the stress–energy tensor. Another one is the orthogonal complement of the metric tensor.
| Property | Value |
|---|---|
| dbo:abstract | A two-vector or bivector is a tensor of type and it is the dual of a two-form, meaning that it is a linear functional which maps two-forms to the real numbers (or more generally, to scalars). The tensor product of a pair of vectors is a two-vector. Then, any two-form can be expressed as a linear combination of tensor products of pairs of vectors, especially a linear combination of tensor products of pairs of basis vectors. If f is a two-vector, then where the f α β are the components of the two-vector. Notice that both indices of the components are contravariant. This is always the case for two-vectors, by definition. A bivector may operate on a one-form, yielding a vector: , although a problem might be which of the upper indices of the bivector to contract with. (This problem does not arise with mixed tensors because only one of such tensor's indices is upper.) However, if the bivector is symmetric then the choice of index to contract with is indifferent. An example of a bivector is the stress–energy tensor. Another one is the orthogonal complement of the metric tensor. (en) |
| dbo:wikiPageID | 2213741 (xsd:integer) |
| dbo:wikiPageLength | 5413 (xsd:nonNegativeInteger) |
| dbo:wikiPageRevisionID | 997793522 (xsd:integer) |
| dbo:wikiPageWikiLink | dbr:Vector_(geometric) dbr:Dyadics dbr:Covariance_and_contravariance_of_vectors dbr:Stress–energy_tensor dbr:Two-point_tensor dbr:Dual_space dbr:Tensor dbr:Tensor_product dbr:Abstract_algebra dbc:Tensors dbr:Bivector dbr:Symmetric_tensor dbr:Metric_tensor dbr:Two-form |
| dct:subject | dbc:Tensors |
| rdf:type | yago:WikicatTensors yago:Abstraction100002137 yago:Cognition100023271 yago:Concept105835747 yago:Content105809192 yago:Idea105833840 yago:PsychologicalFeature100023100 yago:Quantity105855125 yago:Tensor105864481 yago:Variable105857459 |
| rdfs:comment | A two-vector or bivector is a tensor of type and it is the dual of a two-form, meaning that it is a linear functional which maps two-forms to the real numbers (or more generally, to scalars). The tensor product of a pair of vectors is a two-vector. Then, any two-form can be expressed as a linear combination of tensor products of pairs of vectors, especially a linear combination of tensor products of pairs of basis vectors. If f is a two-vector, then , An example of a bivector is the stress–energy tensor. Another one is the orthogonal complement of the metric tensor. (en) |
| rdfs:label | Two-vector (en) |
| owl:sameAs | freebase:Two-vector yago-res:Two-vector wikidata:Two-vector https://global.dbpedia.org/id/4wdoq |
| prov:wasDerivedFrom | wikipedia-en:Two-vector?oldid=997793522&ns=0 |
| foaf:isPrimaryTopicOf | wikipedia-en:Two-vector |
| is foaf:primaryTopic of | wikipedia-en:Two-vector |