Del in cylindrical and spherical coordinates

This is a list of some vector calculus formulae for working with common curvilinear coordinate systems.
Cartesian derivation.
The expressions for formula_14 and formula_15 are found in the same way.
Unit vector conversion formula.
The unit vector of a coordinate parameter "u" is defined in such a way that a small positive change in "u" causes the position vector formula_16 to change in formula_17 direction.
Therefore, 
where "s" is the arc length parameter.
For two sets of coordinate systems formula_19 and formula_20, according to chain rule, 
Now, we isolate the formula_22 component. For formula_23, let formula_24. Then divide on both sides by formula_25 to get: