Skew coordinates

A system of skew coordinates is a curvilinear coordinate system where the coordinate surfaces are not orthogonal,^[1] in contrast to orthogonal coordinates.

Skew coordinates tend to be more complicated to work with compared to orthogonal coordinates since the metric tensor will have nonzero off-diagonal components, preventing many simplifications in formulas for tensor algebra and tensor calculus. The nonzero off-diagonal components of the metric tensor are a direct result of the non-orthogonality of the basis vectors of the coordinates, since by definition:^[2]

${\displaystyle g_{ij}=\mathbf {e} _{i}\cdot \mathbf {e} _{j}}$

where ${\displaystyle g_{ij}}$ is the metric tensor and ${\displaystyle \mathbf {e} _{i}}$ the (covariant) basis vectors.

These coordinate systems can be useful if the geometry of a problem fits well into a skewed system. For example, solving Laplace's equation in a parallelogram will be easiest when done in appropriately skewed coordinates.