Curvilinear coordinates

Any coordinate system built from a smooth map, not necessarily straight axes -- polar, cylindrical, and spherical are the familiar cases.

Curvilinear coordinates replace the rectangular grid with the coordinate curves of a smooth map $\mathbf{r}(u_1,u_2,u_3)$ . Each coordinate curve is generally not a straight line, and each has its own local scale factor and basis vector, both varying from point to point.

Examples

Polar, cylindrical, spherical, parabolic, and elliptic coordinates are all curvilinear systems.