I want ask another basic question related to this paper - http://www.tandfonline.com/doi/pdf/10.1080/16742834.2011.11446922 If I have basis vectors for a curvilinear coordinate system(Euclidean space) that are completely orthogonal to each other(basis vectors will change from point to point). I presume I can have a set of covariant basis vectors and contravariant basis vectors. Now assume that one of those three axes will be non orthogonal to the other two(the paper calls it the sigma coordinate). Will the following statement be correct ? The basis vectors that are orthogonal to each other will transform covariantly and the basis vectors that describe the non orthogonal coordinate surface will transform contravariantly. If the above statement is incorrect can someone explain what this text means from the paper- "In a σ-coordinate, the horizontal covariant basis vectors and the vertical contravariant basis vectors vary in the horizontal and vertical, respectively, while the covariant and contravariant basis vectors are non-orthogonal when the height and slope of terrain do not equal zero "