What is the scale factor in orthogonal vector calculus?

[ElectricSenpai]

Could someone explain to me in simplest of terms what scale factor is when dealing with orthogonal vectors.

 

[andrewkirk]

Given a coordinate system $(x',y',z')$, the scale factor $h_{x'}$ of coordinate $x'$ is

$$\lim_{\delta x'\to 0}\frac{D((x'+\delta x',y'z'),(x',y'z'))}{\delta x'}$$
where $D( (a,b,c),(d,e,f))$ is the distance from point $(a,b,c)$ to point $(d,e,f)$.

In other words, it's the ratio of the size of the displacement to the change in coordinate $x'$ when a tiny increment is added to coordinate $x'$.

Analogous definitions apply for $h_{y'}$ and $h_{z'}$.

Note that the scale factor can change with position. For cylindrical coordinates $h_\phi$ increases with the distance from the $z$ axis.