Abstract
This chapter provides a summary of formulae for the decomposition of a Cartesian second rank tensor into its isotropic, antisymmetric and symmetric traceless parts.
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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:
The dual vector \(\mathbf {c}\) is linked with the antisymmetric part of the tensor by
The symmetric traceless second rank tensor, as defined previously, is
Similarly, for a dyadic tensor composed of the components of the two vectors \(\mathbf {a}\) and \(\mathbf {b}\), the relations above give
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
The symmetric traceless part of the dyadic tensor is
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© 2015 Springer International Publishing Switzerland
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Hess, S. (2015). Summary: Decomposition of Second Rank Tensors. In: Tensors for Physics. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-12787-3_6
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DOI: https://doi.org/10.1007/978-3-319-12787-3_6
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Online ISBN: 978-3-319-12787-3
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