A transport equation such as Ohm’s law (11.74),relates two vectors, which are the conductive electric current density j_q and electric potential gradient ∇φ , through a linear application, which is the electric resistivity ρ. Mathematically, a vector is a rank-1 tensor and a linear application between two vectors is a rank-2 tensor.a) Show that the electric resistivity ρ can be decomposed into the sum of a symmetric part ρs and an antisymmetric part ρa.b) Show that the antisymmetric part ρa has a contribution to the transport that can be written as,where ∇^aφ  is the antisymmetric part of the electric potential gradient and ˆu is a unit axial vector. The decomposition and the expression for the antisymmetric part of the electric potential gradient is a general result that applies for any empirical linear relation between a current density vector and a generalised force vector.

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Vy=- p.jq