▁then ▁ ▁If ▁we ▁set ▁, ▁then ▁since ▁, ▁we ▁have ▁ ▁which ▁provides ▁a ▁means ▁of ▁extract ing ▁the ▁cont rav ari ant ▁component ▁of ▁a ▁vector ▁c . ▁ ▁If ▁bi ▁is ▁the ▁cov ari ant ▁( or ▁natural ) ▁basis ▁at ▁a ▁point , ▁and ▁if ▁bi ▁is ▁the ▁cont rav ari ant ▁( or ▁reci pro cal ) ▁basis ▁at ▁that ▁point , ▁then ▁ ▁A ▁brief ▁r ationale ▁for ▁this ▁choice ▁of ▁basis ▁is ▁given ▁in ▁the ▁next ▁section . ▁ ▁Vector ▁field ▁A ▁similar ▁process ▁can ▁be ▁used ▁to ▁arrive ▁at ▁the ▁gradient ▁of ▁a ▁vector ▁field ▁f ( x ). ▁ ▁The ▁gradient ▁is ▁given ▁by ▁ ▁If ▁we ▁consider ▁the ▁gradient ▁of ▁the ▁position ▁vector ▁field ▁r ( x ) ▁= ▁x , ▁then ▁we ▁can ▁show ▁that ▁ ▁The ▁vector ▁field ▁bi ▁is ▁tang ent ▁to ▁the ▁q i ▁coordinate ▁curve ▁and ▁forms ▁a ▁natural ▁basis ▁at ▁each ▁point ▁on ▁the ▁curve . ▁ ▁This ▁basis , ▁as ▁discussed ▁at ▁the ▁beginning ▁of ▁this ▁article , ▁is ▁also ▁called ▁the ▁cov ari ant ▁cur vil ine ar ▁basis . ▁ ▁We ▁can ▁also ▁define ▁a ▁reci pro cal ▁basis , ▁or ▁cont rav ari ant ▁cur vil ine ar ▁basis , ▁bi . ▁ ▁All ▁the ▁algebraic ▁relations ▁between ▁the ▁basis ▁vectors , ▁as ▁discussed ▁in ▁the ▁section ▁on ▁tensor ▁algebra , ▁apply ▁for ▁the ▁natural ▁basis ▁and ▁its ▁reci pro cal ▁at ▁each ▁point ▁x . ▁ ▁Since ▁c ▁is ▁arbitrary , ▁we