Curvilinear Coordinate Systems Suitable for the Numerical Solution of Boundary-value Problems

The methods of tensor analysis are used to derive a system of non-linear integrability equations associated with a general transformation from Cartesian coordinates to curvilinear coordinates. It is shown that when the curvilinear coordinate system contains one right angle the integrability equations can be integrated and the general form of the transformation can be obtained provided the unit normal to an arbitrarily shaped, smooth, bounded closed surface is prescribed. The general form of the transformation can also be obtained when the unit normal to the boundary section by the longitudinal plane is prescribed. It is then shown how such a system of general curvilinear coordinates can be applied to a large class of boundary value problems in mathematical physics. 1