▁not ing ▁that ▁x ▁= ▁r ( x ). ▁ ▁At ▁each ▁point ▁we ▁can ▁construct ▁a ▁small ▁line ▁element ▁dx . ▁ ▁The ▁square ▁of ▁the ▁length ▁of ▁the ▁line ▁element ▁is ▁the ▁scalar ▁product ▁dx ▁• ▁dx ▁and ▁is ▁called ▁the ▁metric ▁of ▁the ▁space . ▁ ▁Rec all ▁that ▁the ▁space ▁of ▁interest ▁is ▁assumed ▁to ▁be ▁E uclidean ▁when ▁we ▁talk ▁of ▁cur vil ine ar ▁coordinates . ▁Let ▁us ▁express ▁the ▁position ▁vector ▁in ▁terms ▁of ▁the ▁background , ▁fixed , ▁Cart esian ▁basis , ▁i . e ., ▁ ▁Using ▁the ▁chain ▁rule , ▁we ▁can ▁then ▁express ▁dx ▁in ▁terms ▁of ▁three - dimensional ▁orthogonal ▁cur vil ine ar ▁coordinates ▁( q 1 , ▁q 2 , ▁q 3 ) ▁as ▁ ▁Therefore , ▁the ▁metric ▁is ▁given ▁by ▁ ▁The ▁symmetric ▁quantity ▁ ▁is ▁called ▁the ▁fundamental ▁( or ▁metric ) ▁tensor ▁of ▁the ▁E uclidean ▁space ▁in ▁cur vil ine ar ▁coordinates . ▁ ▁Note ▁also ▁that ▁ ▁where ▁hij ▁are ▁the ▁Lam é ▁coefficients . ▁ ▁If ▁we ▁define ▁the ▁scale ▁factors , ▁hi , ▁ ▁using ▁ ▁we ▁get ▁a ▁relation ▁between ▁the ▁fundamental ▁tensor ▁and ▁the ▁Lam é ▁coefficients . ▁ ▁Example : ▁Pol ar ▁coordinates ▁If ▁we ▁consider ▁polar ▁coordinates ▁for ▁R 2 , ▁note ▁that ▁▁ ▁( r , ▁ θ ) ▁are ▁the ▁cur vil ine ar ▁coordinates , ▁and ▁the ▁Jacob ian ▁determin ant ▁of ▁the ▁transformation ▁( r , θ ) ▁→ ▁( r ▁cos ▁