Elements of Classical and Quantum Physics pp 123-133 | Cite as

# Curvilinear Coordinates and Curved Spaces

## Abstract

Even in flat Euclidean space it may be useful to use curvilinear coordinates; for instance, in 3d 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 2d curved subspace embedded in a 3d Euclidean space. The curved coordinates are intrinsic to the surface, and one can ignore the existence of a radial dimension.