Wikipedia gives the relationship between a cartesian and curvlinear coordinate system as gi=(partial)x1/(partial)zi +(partial)x2/(partial)zi http://en.wikipedia.org/wiki/Curvilinear_coordinates Where gi is the i'th basis in the curvlinear coordinate system, x1 and x2 are the cartesian coordinates, zi, is the i'th curvlinear coordinate. Taking i as 1 or 2 in the two dimensional case. I am having trouble deriving a proof for why this is true. Despite being a first year mathematics student i still feel that this is something within my grasp (I am currently studying vector calculus). Could someone please provide me with some geometric intuition for this identity?