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The tensor product between modules A and B is a more general notion than the vector space tensor product. In this case, we replace "scalars" by a ring R. The familiar ...

A covariant tensor, denoted with a lowered index (e.g., a_mu) is a tensor having specific transformation properties. In general, these transformation properties differ from ...

The total number of contravariant and covariant indices of a tensor. The rank R of a tensor is independent of the number of dimensions N of the underlying space. An intuitive ...

Abstractly, the tensor direct product is the same as the vector space tensor product. However, it reflects an approach toward calculation using coordinates, and indices in ...

A contravariant tensor is a tensor having specific transformation properties (cf., a covariant tensor). To examine the transformation properties of a contravariant tensor, ...

An affine tensor is a tensor that corresponds to certain allowable linear coordinate transformations, T:x^_^i=a^i_jx^j, where the determinant of a^i_j is nonzero. This ...

A second-tensor rank symmetric tensor is defined as a tensor A for which A^(mn)=A^(nm). (1) Any tensor can be written as a sum of symmetric and antisymmetric parts A^(mn) = ...

Suppose that V is a group representation of G, and W is a group representation of H. Then the vector space tensor product V tensor W is a group representation of the group ...

The contraction of a tensor is obtained by setting unlike indices equal and summing according to the Einstein summation convention. Contraction reduces the tensor rank by 2. ...

J_(nualphabeta)^mu=J_(nubetaalpha)^mu=1/2(R_(alphanubeta)^mu+R_(betanualpha)^mu), where R is the Riemann tensor.