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.
synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
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 .
For instance section 2.4 of
Last revised on February 3, 2019 at 10:41:30. See the history of this page for a list of all contributions to it.