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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

\end{displaymath} (1)

The Line Element (first Fundamental Form) is then given by
$\displaystyle ds^2$ $\textstyle =$ $\displaystyle g_{11}\,{dx_{11}}^2+g_{22}\,{dx_{22}}^2+g_{33}\,{dx_{33}}^2$ (2)
  $\textstyle =$ $\displaystyle {h_1}^2\,{dx_{11}}^2+{h_2}^2\,{dx_{22}}^2+{h_3}^2\,{dx_{33}}^2.$ (3)

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

See also Curvilinear Coordinates, Fundamental Forms, Line Element

© 1996-9 Eric W. Weisstein