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Curvilinear Coordinate Systems

1. Curvilinear Coordinate Systems are Coordinate Systems in which Atleast One of the Coordinate Axes is a Non Straight Line Curve.
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 may be Orthogonal or Oblique.
4. Curvilinear Coordinate Systems May or May Not have an Origin.
5. 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 Corresponding 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 Corresponding 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 Corresponding 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
   
   \(u=f_u(x,y,z)\hspace{6mm}v=f_v(x,y,z)\hspace{6mm}w=f_w(x,y,z)\)