- order ▁tens ors : ▁The ▁inner ▁product ▁of ▁two ▁second - order ▁tens ors ▁ ▁can ▁be ▁expressed ▁in ▁cur vil ine ar ▁coordinates ▁as ▁ ▁Alternatively , ▁ ▁Det erm in ant ▁of ▁a ▁second - order ▁tensor : ▁If ▁ ▁is ▁a ▁second - order ▁tensor , ▁then ▁the ▁determin ant ▁is ▁defined ▁by ▁the ▁relation ▁ ▁where ▁ ▁are ▁arbitrary ▁vectors ▁and ▁ ▁Rel ations ▁between ▁cur vil ine ar ▁and ▁Cart esian ▁basis ▁vectors ▁Let ▁( e 1 , ▁e 2 , ▁e 3 ) ▁be ▁the ▁usual ▁Cart esian ▁basis ▁vectors ▁for ▁the ▁E uclidean ▁space ▁of ▁interest ▁and ▁let ▁ ▁where ▁F i ▁is ▁a ▁second - order ▁transformation ▁tensor ▁that ▁maps ▁e i ▁to ▁bi . ▁ ▁Then , ▁ ▁From ▁this ▁relation ▁we ▁can ▁show ▁that ▁ ▁Let ▁ ▁be ▁the ▁Jacob ian ▁of ▁the ▁transformation . ▁ ▁Then , ▁from ▁the ▁definition ▁of ▁the ▁determin ant , ▁ ▁Since ▁ ▁we ▁have ▁ ▁A ▁number ▁of ▁interesting ▁results ▁can ▁be ▁derived ▁using ▁the ▁above ▁relations . ▁ ▁First , ▁consider ▁ ▁Then ▁ ▁Similarly , ▁we ▁can ▁show ▁that ▁ ▁Therefore , ▁using ▁the ▁fact ▁that ▁, ▁ ▁Another ▁interesting ▁relation ▁is ▁derived ▁below . ▁ ▁Rec all ▁that ▁ ▁where ▁A ▁is ▁a , ▁yet ▁und et erm ined , ▁constant . ▁ ▁Then ▁ ▁This ▁observation ▁leads ▁to ▁the ▁relations ▁ ▁In ▁index ▁notation , ▁ ▁where ▁ ▁is ▁the ▁usual ▁permut ation ▁symbol . ▁ ▁We ▁have ▁not ▁identified ▁an