Curvilinear Co-ordinate Systems Associated with a Class of Boundary-Value Problems

A system of non-linear integrability equations is derived whichis associated with the differential form of a transformationfrom Cartesian co-ordinates to non-orthogonal curvilinear co-ordinates.A solution for this system is established when the curvilinearco-ordinate system contains two identical scaling factors andone right angle provided the unit normal to a smooth, finiteor infinite tube is prescribed. The general form of the transformationis obtained. It is shown that the transformation can also beobtained when the unit normal to the tube boundary section bythe plane normal to the given curve defining the orientationof the tube is prescribed. Moreover conditions are establishedunder which a completely orthogonal co-ordinate system can befound. An example is treated for a tube with circular cross-sectionand both the non-orthogonal and orthogonal co-ordinate systemsassociated with this are discussed.