Value of Curvilinear Coordinate Metric

Definition
Let $A$ and $B$ be two points in space.

Let a curvilinear $3$-Space coordinate system $\QQ$ be applied on top of a Cartesian $3$-space.

Let $h_{i j}$ be the metric of $\QQ$.

Then:
 * $\forall i, j \in \set {1, 2, 3}: {h_{i j} }^2 = \dfrac {\partial x} {\partial q_i} \dfrac {\partial x} {\partial q_j} + \dfrac {\partial y} {\partial q_i} \dfrac {\partial y} {\partial q_j} + \dfrac {\partial z} {\partial q_i} \dfrac {\partial z} {\partial q_j}$

where $\partial$ denotes partial differentiation.

Proof
From the Cartesian representation of $\QQ$:

where:
 * $\tuple {x, y, z}$ denotes the Cartesian coordinates


 * $\tuple {q_1, q_2, q_3}$ denotes the corresponding curvilinear coordinates.

Then:

By definition of the metric of $\QQ$:

Thus we can extract the appropriate terms with $\d q_i \rd q_j$:

Hence the result.