**Example 17.10.13 [Polar coordinates in space]** The standard polar coordinates in space (§7.1) are obtained by rotating the plane polar coordinates around the *x*-axis. From (17.10.48) and (17.10.56) we have

(17.10.58) |

and from (17.10.57) *h*_{X} = *h*_{Y} = *e*^{ξ}, *h*_{Z} = *e*^{ξ} cos *η*. The substitution *r* = *e*^{ξ} reduces these expressions to the usual form.

Rotation of the plane bi-polar coordinate system around the *x*-axis generates bi-spherical coordinates in which the coordinate surfaces are spheres, spindles and half-planes. Rotation around the *y*-axis generates toroidal coordinates in which the coordinate surfaces are tori and intersecting spheres.

This ...

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