This is about tensor quantities in the sense of multilinear algebra, differential geometry and physics, as in tensor calculus. For the different notion of a tensor in enriched category theory see under copower.
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Generally, a tensor is an element of a tensor product.
Traditionally this is considered in differential geometry for the following case:
for a manifold, the tangent bundle, the cotangent bundle, , their spaces of sections and the associative algebra of functions on , a rank- tensor or tensor field on is an element of the tensor product of modules over
A rank -tensor is also called a covariant tensor and a rank -tensor a contravariant tensor.
(…)
A vector field is a rank -tensor field.
A Riemannian metric is a symmetric rank -tensor.
A differential form of degree is a skew-symmetric rank -tensor.
A Poisson tensor is a skew-symmetric tensor of rank .
Historical discussion:
Monograph:
Discussion with an eye towards application in (particle) physics:
Theodore Frankel, section 2.4 in: The Geometry of Physics - An Introduction
Howard Georgi, §10 in: Lie Algebras In Particle Physics, Westview Press (1999), CRC Press (2019) [doi:10.1201/9780429499210]
Last revised on May 24, 2024 at 10:19:41. See the history of this page for a list of all contributions to it.