Curvilinear Coordinates and Curved Spaces

Even in flat Euclidean space, it may be useful to use curvilinear coordinates; for instance, in three-dimensional problems having central symmetry, we obtain an important simplification when the line element \(ds^2= dx^{2}+dy^{2}+dz^{2}\) is replaced by \(ds^{2}=dr^{2}+r^{2}d\theta ^{2}+r^{2}\sin ^{2}(\theta ) d\phi ^{2}.\)  In a curved space we have no other choice, because Cartesian coordinates may exist only locally, that is, in an infinitesimal neighborhood. One example is the surface of a sphere of radius R. The spherical coordinates with r set equal to the radius R  of the sphere do the job. In the latter case, we deal with a two-dimensional curved subspace embedded in a three-dimensional Euclidean space. The curved coordinates are intrinsic to the surface, and one can ignore the existence of a radial dimension.