nLab
tensor

This is about tensor quantities in the sense of multilinear algebra, differential geometry and physics. For the different notion of a tensor in enriched category theory see under copower.


Context

Category theory

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Definition

Generally, a tensor is an element of a tensor product.

Traditionally this is considered in differential geometry for the following case:

for XX a manifold, TXT X the tangent bundle, T *XT^* X the cotangent bundle, Γ(TX)\Gamma(T X), Γ(T *X)\Gamma(T^* X) their spaces of sections and C(X)C(X) the associative algebra of functions on XX, a rank-(p,q)(p,q) tensor or tensor field on XX is an element of the tensor product of modules over C(X)C(X)

tΓ(TX) C(X) p C(X)Γ(T *X) C(X) q. t \in \Gamma(T X)^{\otimes_{C(X)}^p} \otimes_{C(X)} \Gamma(T^* X)^{\otimes^q_{C(X)}} \,.

A rank (p,0)(p,0)-tensor is also called a covariant tensor and a rank (0,q)(0,q)-tensor a contravariant tensor.

Examples

General

(…)

In differential geometry

References

For instance section 2.4 of

Revised on June 21, 2013 01:48:34 by Urs Schreiber (82.169.65.155)