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  
\[ds^2 = d \mathbf{r} \cdot d \mathbf{r}\]

\[d \mathbf{r}=\frac{\partial d \mathbf{r}}{\partial \alpha} d \alpha +\frac{\partial d \mathbf{r}}{\partial \beta} d \beta\]
\[ \begin{equation} \begin{aligned} ds^2 &= d \mathbf{r} \cdot d \mathbf{r} \\ &=(\frac{\partial d \mathbf{r}}{\partial \alpha} d \alpha +\frac{\partial d \mathbf{r}}{\partial \beta} d \beta) \cdot (\frac{\partial d \mathbf{r}}{\partial \alpha} d \alpha +\frac{\partial d \mathbf{r}}{\partial \beta} d \beta) \\ &=\frac{\partial \mathbf{r}}{\partial \alpha} \cdot \frac{\partial \mathbf{r}}{\partial \alpha} d \alpha^2 +2 \frac{\partial \mathbf{r}}{\partial \alpha} \cdot \frac{\partial \mathbf{r}}{\partial \beta} d \alpha d \beta +\frac{\partial \mathbf{r}}{\partial \beta} \cdot \frac{\partial \mathbf{r}}{\partial \beta} d \beta^2 \end{aligned} \end{equation}\]
If the curvilinear coordinates are orthogonal then the curves in the surface
\[\frac{\partial \mathbf{r}}{\partial \alpha}\]
\[\frac{\partial \mathbf{r}}{\partial \beta}\]
  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.

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