▁orth on ormal ▁right - hand ed ▁basis , ▁the ▁third - order ▁altern ating ▁tensor ▁is ▁defined ▁as ▁ ▁In ▁a ▁general ▁cur vil ine ar ▁basis ▁the ▁same ▁tensor ▁may ▁be ▁expressed ▁as ▁ ▁It ▁can ▁be ▁shown ▁that ▁ ▁Now , ▁ ▁Hence , ▁ ▁Similarly , ▁we ▁can ▁show ▁that ▁ ▁Vector ▁operations ▁ ▁Identity ▁map ▁The ▁identity ▁map ▁I ▁defined ▁by ▁ ▁can ▁be ▁shown ▁to ▁be ▁ ▁S cal ar ▁( dot ) ▁product ▁The ▁scalar ▁product ▁of ▁two ▁vectors ▁in ▁cur vil ine ar ▁coordinates ▁is ▁ ▁Vector ▁( cross ) ▁product ▁The ▁cross ▁product ▁of ▁two ▁vectors ▁is ▁given ▁by ▁ ▁where ▁ ε ijk ▁is ▁the ▁permut ation ▁symbol ▁and ▁e i ▁is ▁a ▁Cart esian ▁basis ▁vector . ▁ ▁In ▁cur vil ine ar ▁coordinates , ▁the ▁equivalent ▁expression ▁is ▁ ▁where ▁ ▁is ▁the ▁third - order ▁altern ating ▁tensor . ▁The ▁cross ▁product ▁of ▁two ▁vectors ▁is ▁given ▁by ▁ ▁where ▁ ε ijk ▁is ▁the ▁permut ation ▁symbol ▁and ▁ ▁is ▁a ▁Cart esian ▁basis ▁vector . ▁ ▁Therefore , ▁ ▁and ▁ ▁Hence , ▁ ▁Return ing ▁to ▁the ▁vector ▁product ▁and ▁using ▁the ▁relations ▁ ▁gives ▁us ▁ ▁T ensor ▁operations ▁ ▁Identity ▁map : ▁The ▁identity ▁map ▁ ▁defined ▁by ▁ ▁can ▁be ▁shown ▁to ▁be ▁ ▁Action ▁of ▁a ▁second - order ▁tensor ▁on ▁a ▁vector : ▁The ▁action ▁ ▁can ▁be ▁expressed ▁in ▁cur vil ine ar ▁coordinates ▁as ▁ ▁Inner ▁product ▁of ▁two ▁second