Definition:Metric (Curvilinear Coordinates)

Definition

Let $S$ be an infinitesimal arc.

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

Then:

\(\ds \d s^2\)

\(=\)

\(\ds \d x^2 + \d y^2 + \d z^2\)

\(\ds \)

\(=\)

\(\ds \sum_{i, j} {h_{i j} }^2 \rd q_i \rd q_j\)

for $i, j \in \set {1, 2, 3}$

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 $\QQ$.

Sources

• 1970: George Arfken: Mathematical Methods for Physicists (2nd ed.) ... (previous) ... (next): Chapter $2$ Coordinate Systems $2.1$ Curvilinear Coordinates: $(2.4)$