Scale Factor

For a diagonal Metric Tensor $g_{ij}=g_{ii}\delta_{ij}$, where $\delta_{ij}$ is the Kronecker Delta, the scale factor is defined by

$$h_i\equiv\sqrt{g_{ii}}.$$ (1)

The Line Element (first Fundamental Form) is then given by

$$ds^2 = g_{11}\,{dx_{11}}^2+g_{22}\,{dx_{22}}^2+g_{33}\,{dx_{33}}^2$$ (2)

$$= {h_1}^2\,{dx_{11}}^2+{h_2}^2\,{dx_{22}}^2+{h_3}^2\,{dx_{33}}^2.$$ (3)

The scale factor appears in vector derivatives of coordinates in Curvilinear Coordinates.

See also Curvilinear Coordinates, Fundamental Forms, Line Element