▁can ▁write ▁ ▁Note ▁that ▁the ▁cont rav ari ant ▁basis ▁vector ▁bi ▁is ▁per pend icular ▁to ▁the ▁surface ▁of ▁constant ▁ ψ i ▁and ▁is ▁given ▁by ▁ ▁Christ off el ▁symbols ▁of ▁the ▁first ▁kind ▁The ▁Christ off el ▁symbols ▁of ▁the ▁first ▁kind ▁are ▁defined ▁as ▁ ▁To ▁express ▁ Γ ijk ▁in ▁terms ▁of ▁g ij ▁we ▁note ▁that ▁ ▁Since ▁bi , j ▁= ▁b j , i ▁we ▁have ▁ Γ ijk ▁= ▁ Γ j ik . ▁ ▁Using ▁these ▁to ▁rear range ▁the ▁above ▁relations ▁gives ▁ ▁Christ off el ▁symbols ▁of ▁the ▁second ▁kind ▁The ▁Christ off el ▁symbols ▁of ▁the ▁second ▁kind ▁are ▁defined ▁as ▁ ▁in ▁which ▁ ▁This ▁implies ▁that ▁ ▁Other ▁relations ▁that ▁follow ▁are ▁ ▁Another ▁particularly ▁useful ▁relation , ▁which ▁shows ▁that ▁the ▁Christ off el ▁symbol ▁depends ▁only ▁on ▁the ▁metric ▁tensor ▁and ▁its ▁derivatives , ▁is ▁ ▁Exp licit ▁expression ▁for ▁the ▁gradient ▁of ▁a ▁vector ▁field ▁The ▁following ▁expressions ▁for ▁the ▁gradient ▁of ▁a ▁vector ▁field ▁in ▁cur vil ine ar ▁coordinates ▁are ▁quite ▁useful . ▁ ▁Represent ing ▁a ▁physical ▁vector ▁field ▁The ▁vector ▁field ▁v ▁can ▁be ▁represented ▁as ▁ ▁where ▁ ▁are ▁the ▁cov ari ant ▁components ▁of ▁the ▁field , ▁ ▁are ▁the ▁physical ▁components , ▁and ▁( no ▁summ ation ) ▁ ▁is ▁the ▁normal ized ▁cont rav ari ant ▁basis ▁vector . ▁ ▁Second - order ▁tensor ▁field ▁The ▁gradient ▁of ▁a ▁second ▁order ▁tensor ▁field ▁can ▁similarly