Abstract: Though the transformation matrix between the unit basic vector in deferent orthogonal curvilinear coordinate system, we deduce the law of the unit basic vectors derivative of their own coordinates and the time in the orthogonal curvilinear coordinate system. As an example, the derivation of the unit basic vectors self space coordinate in the spherical coordinate system and the cylindrical coordinates system is also discussed.

Key words: orthogonal curvilinear coordinate system, unitary matrix, unit basic vectors, derivative