Skew coordinates

A system of skew coordinates is a curvilinear coordinate system where the coordinate surfaces are not orthogonal, in contrast to orthogonal 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: where g i j {displaystyle g_{ij}} is the metric tensor and e i {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. The simplest 3D case of a skew coordinate system is a Cartesian one where one of the axes (say the x axis) has been bent by some angle ϕ {displaystyle phi } , staying orthogonal to one of the remaining two axes. For this example, the x axis of a Cartesian coordinate has been bent toward the z axis by ϕ {displaystyle phi } , remaining orthogonal to the y axis. Let e 1 {displaystyle mathbf {e} _{1}} , e 2 {displaystyle mathbf {e} _{2}} , and e 3 {displaystyle mathbf {e} _{3}} respectively be unit vectors along the x {displaystyle x} , y {displaystyle y} , and z {displaystyle z} axes. These represent the covariant basis; computing their dot products gives the following components of the metric tensor: