dimensional ▁E uclidean ▁space . ▁ ▁In ▁general , ▁the ▁basis ▁vectors ▁are ▁neither ▁unit ▁vectors ▁nor ▁mut ually ▁orthogonal . ▁ ▁However , ▁they ▁are ▁required ▁to ▁be ▁linear ly ▁independent . ▁ ▁Then ▁a ▁vector ▁v ▁can ▁be ▁expressed ▁as ▁ ▁The ▁components ▁v k ▁are ▁the ▁cont rav ari ant ▁components ▁of ▁the ▁vector ▁v . ▁ ▁The ▁reci pro cal ▁basis ▁( b 1 , ▁b 2 , ▁b 3 ) ▁is ▁defined ▁by ▁the ▁relation ▁▁ ▁where ▁ δ i ▁j ▁is ▁the ▁Kr one cker ▁delta . ▁ ▁The ▁vector ▁v ▁can ▁also ▁be ▁expressed ▁in ▁terms ▁of ▁the ▁reci pro cal ▁basis : ▁ ▁The ▁components ▁v k ▁are ▁the ▁cov ari ant ▁components ▁of ▁the ▁vector ▁. ▁ ▁Second - order ▁tens ors ▁in ▁cur vil ine ar ▁coordinates ▁A ▁second - order ▁tensor ▁can ▁be ▁expressed ▁as ▁ ▁The ▁components ▁S ij ▁are ▁called ▁the ▁cont rav ari ant ▁components , ▁Si ▁j ▁the ▁mixed ▁right - cov ari ant ▁components , ▁Si ▁j ▁the ▁mixed ▁left - cov ari ant ▁components , ▁and ▁S ij ▁the ▁cov ari ant ▁components ▁of ▁the ▁second - order ▁tensor . ▁ ▁Met ric ▁tensor ▁and ▁relations ▁between ▁components ▁The ▁quantities ▁g ij , ▁g ij ▁are ▁defined ▁as ▁ ▁From ▁the ▁above ▁equations ▁we ▁have ▁ ▁The ▁components ▁of ▁a ▁vector ▁are ▁related ▁by ▁ ▁Also , ▁ ▁The ▁components ▁of ▁the ▁second - order ▁tensor ▁are ▁related ▁by ▁ ▁The ▁altern ating ▁tensor ▁In ▁an