Skew coordinates

A system of skew coordinates is a curvilinear coordinate system where the coordinate surfaces are not orthogonal, 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:

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

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.