Is there any difference between resultant reaction force in Cartesian and cylindrical coordinate system, meaning if I am calculate the resultant force on a ring in Cartesian system and then calculate in cylindrical coordinate system is there any difference between the resultant value??????????? Please explain.....

From Wikipedia:

A

Cartesian coordinate systemspecifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length. http://en.wikipedia.org/wiki/Cartesian_coordinate_systemA

Cylindrical coordinate systemis a three-dimensional coordinate system, where each point is specified by the two polar coordinates of its perpendicular projection onto some fixed plane, and by its (signed) distance from that plane. http://en.wikipedia.org/wiki/Cylindrical_coordinate_systemHence your answer values should be different.