Necessary and Sufficient Conditions for Curvilinear Coordinates to be Orthogonal
\[\mathbf{r}=\mathbf(\alpha , \beta )\]
, 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
\[ \begin{equation} \begin{aligned} ds^2 &= d \mathbf{r} \cdot d \mathbf{r} \\ &=(\frac{\partial d \mathbf{r}}{\partial \alpha} d \alpha +\frac{\partial d \mathbf{r}}{\partial \beta} d \beta) \cdot (\frac{\partial d \mathbf{r}}{\partial \alpha} d \alpha +\frac{\partial d \mathbf{r}}{\partial \beta} d \beta) \\ &=\frac{\partial \mathbf{r}}{\partial \alpha} \cdot \frac{\partial \mathbf{r}}{\partial \alpha} d \alpha^2 +2 \frac{\partial \mathbf{r}}{\partial \alpha} \cdot \frac{\partial \mathbf{r}}{\partial \beta} d \alpha d \beta +\frac{\partial \mathbf{r}}{\partial \beta} \cdot \frac{\partial \mathbf{r}}{\partial \beta} d \beta^2 \end{aligned} \end{equation}\]
.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.
