θ , ▁r ▁sin ▁ θ ) ▁is ▁r . ▁ ▁The ▁orthogonal ▁basis ▁vectors ▁are ▁br ▁= ▁( cos ▁ θ , ▁sin ▁ θ ), ▁b θ ▁= ▁( − r ▁sin ▁ θ , ▁r ▁cos ▁ θ ). ▁ ▁The ▁normal ized ▁basis ▁vectors ▁are ▁er ▁= ▁( cos ▁ θ , ▁sin ▁ θ ), ▁e θ ▁= ▁( − sin ▁ θ , ▁cos ▁ θ ) ▁and ▁the ▁scale ▁factors ▁are ▁hr ▁= ▁ 1 ▁and ▁h θ = ▁r . ▁The ▁fundamental ▁tensor ▁is ▁g 1 1 ▁= 1 , ▁g 2 2 ▁= r 2 , ▁g 1 2 ▁= ▁g 2 1 ▁= 0 . ▁ ▁Line ▁and ▁surface ▁integr als ▁If ▁we ▁wish ▁to ▁use ▁cur vil ine ar ▁coordinates ▁for ▁vector ▁calculus ▁calculations , ▁adjust ments ▁need ▁to ▁be ▁made ▁in ▁the ▁calculation ▁of ▁line , ▁surface ▁and ▁volume ▁integr als . ▁ ▁For ▁simplicity , ▁we ▁again ▁restrict ▁the ▁discussion ▁to ▁three ▁dimensions ▁and ▁orthogonal ▁cur vil ine ar ▁coordinates . ▁ ▁However , ▁the ▁same ▁arguments ▁apply ▁for ▁- dimensional ▁problems ▁though ▁there ▁are ▁some ▁additional ▁terms ▁in ▁the ▁expressions ▁when ▁the ▁coordinate ▁system ▁is ▁not ▁orthogonal . ▁ ▁Line ▁integr als ▁Norm ally ▁in ▁the ▁calculation ▁of ▁line ▁integr als ▁we ▁are ▁interested ▁in ▁calculating ▁▁ ▁where ▁x ( t ) ▁paramet riz es ▁C ▁in ▁Cart esian ▁coordinates . ▁In ▁cur vil ine ar ▁coordinates , ▁the ▁term ▁▁ ▁by ▁the ▁chain ▁rule . ▁ ▁And ▁from