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An aspect of vector variables in curvilinear coordinates that is sometimes overlooked is that the derivative of a vector is not necessarily the same as the vector of derivatives of the components. This is because in differentiating a vector, the unit vectors in the coordinate space must also be differentiated.
In cylindrical (R,Phi,Z) coordinates, for example, the radial component of the Laplacian of a vector V is
DEL2(Vr)  Vr/R^2  2*DPHI(Vphi)/R^2
The extra 1/R^2 terms have arisen from the differentiation of the unit vectors.
FlexPDE performs the correct expansion of the differential operators in all supported coordinate systems.