## Necessary and Sufficient Conditions for Curvilinear Coordinates to be Orthogonal

Given a surface
$\mathbf{r}=\mathbf(\alpha , \beta )$
, the differential for the arc length is given by the formula
$ds^2 = d \mathbf{r} \cdot d \mathbf{r}$

Where
$d \mathbf{r}=\frac{\partial d \mathbf{r}}{\partial \alpha} d \alpha +\frac{\partial d \mathbf{r}}{\partial \beta} d \beta$
,
Hence
\begin{aligned} ds^2 &= d \mathbf{r} \cdot d \mathbf{r} \\ &=(\frac{\partial d \mathbf{r}}{\partial \alpha} d \alpha +\frac{\partial d \mathbf{r}}{\partial \beta} d \beta) \cdot (\frac{\partial d \mathbf{r}}{\partial \alpha} d \alpha +\frac{\partial d \mathbf{r}}{\partial \beta} d \beta) \\ &=\frac{\partial \mathbf{r}}{\partial \alpha} \cdot \frac{\partial \mathbf{r}}{\partial \alpha} d \alpha^2 +2 \frac{\partial \mathbf{r}}{\partial \alpha} \cdot \frac{\partial \mathbf{r}}{\partial \beta} d \alpha d \beta +\frac{\partial \mathbf{r}}{\partial \beta} \cdot \frac{\partial \mathbf{r}}{\partial \beta} d \beta^2 \end{aligned}
.
If the curvilinear coordinates are orthogonal then the curves in the surface
$\alpha=C_1$
,
$\beta=C_2$
so
$\frac{\partial \mathbf{r}}{\partial \alpha}$
and
$\frac{\partial \mathbf{r}}{\partial \beta}$
are perpendicular and
$\frac{\partial \mathbf{r}}{\partial \alpha} \cdot \frac{\partial \mathbf{r}}{\partial \beta} =0$
.
This is necessary and sufficient for a curvilinear system to be orthogonal.