Tensorwert

Tensorwert is the coordinate representation of a tensor with respect to a chosen basis. In mathematics and physics, a tensor is an abstract multilinear object, while its tensorwert—its components—are the numbers that express the tensor in a given basis. The tensorwert depends on the basis, whereas the tensor itself is an invariant, basis-independent object.

A tensor T of type (r,s) has components T^{i1...ir}_{j1...js} in a basis {e_i}. Upper indices (contravariant) correspond to basis vectors, while lower indices (covariant) correspond to dual basis functionals. Examples include a vector with components v^i, a covector with components v_i, and a rank-2 tensor with components such as T^{ij} or T^{i}_{j} depending on the chosen index placement.

Under a change of basis, the tensorwert transform according to tensor transformation laws. If A is the change-of-basis matrix, the transformed components T'^{i1...ir}_{j1...js} are obtained from T^{p1...pr}_{q1...qs} by multiplying with A for each upper index and with A^{-1} for each lower index, in the appropriate positions. These rules ensure that the same abstract tensor remains invariant even though its components differ between bases.

Matrices can be viewed as the tensorwert of a rank-2 tensor representing a linear map in a given basis; the components form the matrix elements of that map. The metric tensor g_{ij} is a well-known example whose tensorwert encodes distances and angles in a coordinate system.

Terminology note: in German-language texts, Komponenten eines Tensors is the common phrasing; Tensorwert is a direct, less frequent synonym for the same concept.