Tensor field

A tensor field (also imprecise tensor) is examined in the mathematical sub-area of differential geometry, in particular in tensor analysis. It is a function that assigns a tensor to each point of an underlying space in a special way .

definition

Let be a smooth manifold and an (r, s) - tensor bundle . An (r, s) -tensor field is a smooth cut in the tensor bundle . The number of tensor fields is denoted by. This set is a module on the algebra of smooth functions . ${\displaystyle M}$ ${\displaystyle T_{s}^{r}(M)}$ ${\displaystyle T_{s}^{r}(M)}$ ${\displaystyle \Gamma ^{\infty }(T_{s}^{r}(M))}$ ${\displaystyle C^{\infty }(M)=\Gamma ^{\infty }(T_{0}^{0}(M))}$

Examples

Let M be a differentiable manifold, then a tensor field on M is a mapping that assigns a tensor to every point.

• Riemannian metrics are (0.2) tensor fields.
• The Riemann curvature tensor is a (1,3) -tensor field, which can be interpreted as a (0,4) -tensor field with the help of the Riemannian metric.
• Differential forms of degree k, in particular the total differential of a function in the case k = 1, are sections of where the cotangential bundle is called . For more information, see also Outer Algebra . ${\displaystyle \textstyle \bigwedge ^{k}\mathrm {T} ^{*}M\subseteq (\mathrm {T} ^{*}M)^{\otimes k}.}$ ${\displaystyle T^{*}M}$
• The energy-momentum tensor and the electromagnetic field strength tensor (as an example of a field strength tensor) in the theory of relativity are tensor fields of the second order on the four-dimensional basis of Minkowski space . ${\displaystyle T^{\alpha \beta }}$ ${\displaystyle F^{\alpha \beta }}$
• the spin group , whose representations are the often used spinor fields , is usually constructed as a subset of the tensor fields with values ​​in the Clifford algebra .

See also

• Recurrent tensor

source

• R. Abraham, JE Marsden, T. Ratiu: Manifolds, Tensor Analysis, and Applications (= Applied Mathematical Sciences 75). 2nd edition. Springer-Verlag, New York NY et al. 1988, ISBN 0-387-96790-7 .

Web links

Commons : Tensor fields  - collection of images, videos and audio files

• Hendrik van Hees: Physics FAQ for the German-language physics newsgroups http://theory.gsi.de/~vanhees/faq/geo/node10.html