Keywords

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Any second rank tensor \(A_{\mu \nu }\) can be decomposed into its isotropic part, associated with a scalar, its antisymmetric part, linked a vector, and its irreducible, symmetric traceless part:

(6.1)

The dual vector \(\mathbf {c}\) is linked with the antisymmetric part of the tensor by

$$\begin{aligned} c_{\lambda } = \varepsilon _{\lambda \sigma \tau }A_{\sigma \tau } = \varepsilon _{\lambda \sigma \tau }\,\frac{1}{2}\,(A_{\sigma \tau }-A_{\tau \sigma }). \end{aligned}$$
(6.2)

The symmetric traceless second rank tensor, as defined previously, is

(6.3)

Similarly, for a dyadic tensor composed of the components of the two vectors \(\mathbf {a}\) and \(\mathbf {b}\), the relations above give

(6.4)

The isotropic part involves the scalar product \((\mathbf {a}\cdot \mathbf {b})\) of the two vectors. The antisymmetric part is linked with the cross product of the two vectors, here one has

$$\begin{aligned} c_{\lambda } = \varepsilon _{\lambda \sigma \tau }a_{\sigma }b_{\tau } = (\mathbf {a}\times \mathbf {b})_{\lambda }. \end{aligned}$$
(6.5)

The symmetric traceless part of the dyadic tensor is

(6.6)