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