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The trace of a second-tensor rank tensor T is a scalar given by the contracted mixed tensor equal to T_i^i. The trace satisfies ...

A Cartesian tensor is a tensor in three-dimensional Euclidean space. Unlike general tensors, there is no distinction between covariant and contravariant indices for Cartesian ...

G_(ab)=R_(ab)-1/2Rg_(ab), where R_(ab) is the Ricci curvature tensor, R is the scalar curvature, and g_(ab) is the metric tensor. (Wald 1984, pp. 40-41). It satisfies ...

The permutation tensor, also called the Levi-Civita tensor or isotropic tensor of rank 3 (Goldstein 1980, p. 172), is a pseudotensor which is antisymmetric under the ...

A tensor g whose discriminant satisfies g=g_(11)g_(22)-g_(12)^2>0.

The vector space tensor product V tensor W of two group representations of a group G is also a representation of G. An element g of G acts on a basis element v tensor w by ...

Given an antisymmetric second tensor rank tensor C_(ij), a dual pseudotensor C_i is defined by C_i=1/2epsilon_(ijk)C_(jk), (1) where C_i = [C_(23); C_(31); C_(12)] (2) C_(jk) ...

The tensor product of two vector spaces V and W, denoted V tensor W and also called the tensor direct product, is a way of creating a new vector space analogous to ...

A tensor defined in terms of the tensors which satisfy the double contraction relation.

Given a general second tensor rank tensor A_(ij) and a metric g_(ij), define theta = A_(ij)g^(ij)=A_i^i (1) omega^i = epsilon^(ijk)A_(jk) (2) sigma_(ij) = ...