 The mapping that transforms from spherical coordinates to rectangular coordinates is

Using the chain rule again shows that formula_63 All dot products formula_64 where formula_65 and formula_66 differ are zero, so the squared norm of this vector is

So for a curve expressed in spherical coordinates, the arc length is

A very similar calculation shows that the arc length of a curve expressed in cylindrical coordinates is

Arc lengths are denoted by "s", since the Latin word for length (or size) is "spatium".In the following lines, formula_70 represents the radius of a circle, formula_71 is its diameter, formula_72 is its circumference, formula_73 is the length of an arc of the circle, and formula_59 is the angle which the arc subtends at the centre of the circle.The distances formula_75 and formula_73 are expressed in the same units.