Tensor field

A tensor ( also imprecise tensor ) is studied in the mathematical branch of differential geometry in particular in tensor analysis. It is a function that assigns a tensor in a special way each point of the underlying space.

Definition

Let be a smooth manifold and a (r, s) - Tensorbündel. A (r, s) - tensor field is a smooth cut in Tensorbündel. The amount of tensor fields is denoted by. This quantity is a module over the algebra of smooth functions.

Examples

Let M be a differentiable manifold, then M is a tensor field on a mapping which assigns to each point of a tensor.

  • Riemannian metrics are (0,2) - tensor fields.
  • The Riemann curvature tensor is a ( 1,3)- tensor field, which can be interpreted using the Riemannian metric as a (0,4 ) - tensor field.
  • Differential forms of degree k, in particular, the total differential of a function in the case of k = 1, this indicates the sections of cotangent bundle. For more information see after even under exterior algebra.
  • The energy -momentum tensor of the electromagnetic field tensor and ( as an example of Feldstärketensors ) in the theory of relativity are second order tensor fields on the four-dimensional base of the Minkowski space.
  • The spin group whose representations are the spinor fields often used, is usually constructed as a subset of the tensor 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, among others, 1988, ISBN 0-387-96790-7.
Differentialform Stress–energy tensor Clifford-Algebra
765368
de