Definition:Metric (Curvilinear Coordinates)

Definition
Let $S$ be an infinitesimal arc.

Let $\d s$ be the length of $S$

Then:

where:
 * $\d x$, $\d y$ and $\d z$ are the lengths of the projections of $S$ on the $x$-axis, $y$-axis and $z$-axis respectively


 * $\d q_i$ is the projection of $S$ onto the unit normal to the curvilinear coordinate surface determined by $q_i$, for $i \in \set {1, 3}$


 * the coefficients $h_{i j}$ specify the nature of the curvilinear coordinate system $\QQ$ being utilised.

These coefficients $h_{i j}$ are collectively referred to as the metric of $\QQ$.