Curvilinear Coordinate Systems

1. Curvilinear Coordinate Systems are either Oblique Coordinate Systems or are Coordinate Systems in which atleast One of the Coordinate Axes is Not a Straight Line.
2. Unlike Cartesian Coordinate Systems where the Nomenclature of Coordinates / Coordinate Axes are fixed, the Nomenclature of Coordinates / Coordinate Axes in Curvilinear Coordinate Systems vary from one to another depending on the Type of the Curvilinear Coordinate Systems.
3. Curvilinear Coordinate Systems without a well defined Origin are called Affine Coordinate Systems.
4. Any Curvilinear Coordinate System of a given Dimension (with a well defined Origin) has a one to one correspondence with the Cartesian Coordinate System of similar Dimension. That means any Coordinate Location given by the Coordinates of the Curvilinear Coordinate System of a given Dimension can be converted to a Coordinate Location given by the Coordinates of the Cartesian Coordinate System of similar Dimension and vice versa.
   
   For example, given a Coordinate Location \((u, v, w)\) in a 3 Dimensional Curvilinear Coordinate System having Coordinate Axes \(U\) , \(V\) and \(W\), the Coordinate Location in 3 Dimensional Cartesian Coordinate System can be obtained by a set of functions as follows
   
   \(x=f_x(u,v,w)\hspace{6mm}y=f_y(u,v,w)\hspace{6mm}z=f_z(u,v,w)\)
   
   Similarly, given a Coordinate Location \((x, y, z)\) in 3 Dimensional Cartesian Coordinate System, the Coordinate Location \((u, v, w)\) in a 3 Dimensional Curvilinear Coordinate System having Coordinate Axes \(U\) , \(V\) and \(W\) can be obtained by a set of functions as follows
   
   \(u=f_u(x,y,z)\hspace{6mm}v=f_v(x,y,z)\hspace{6mm}w=f_w(x,y,z)\)