Scale factor

The physical length of a unit step in one curvilinear coordinate -- $h_i = \|\partial\mathbf{r}/\partial u_i\|$ .

A coordinate system's scale factor $h_i$ converts a bare coordinate differential $du_i$ into an actual physical length $h_i\,du_i$ . It's what makes the gradient, divergence, curl, and Laplacian formulas in curvilinear coordinates come out correct once $u_i$ no longer directly measures distance.

Examples

In cylindrical coordinates $(r,\theta,z)$ , the scale factors are $h_r=1$ , $h_\theta=r$ , $h_z=1$ .
In spherical coordinates $(\rho,\varphi,\theta)$ , the scale factors are $h_\rho=1$ , $h_\varphi=\rho$ , $h_\theta=\rho\sin\varphi$ .