Element of Arc Length in Terms of Intrinsic Coordinates on a Surface
\[u,v\]
, \[\mathbf{r} =\mathbf{r}(u,v)\]
.Let
\[ds\]
be an element of arc length in the surface.Then
\[d \mathbf{r} = \frac{\partial \mathbf{r}}{\partial u} du + \frac{\partial \mathbf{r}}{\partial v} dv \]
.Hence
\[\begin{equation} \begin{aligned} ds &= \sqrt{d \mathbf{r} \cdot d \mathbf{r}} \\&= \sqrt{(\frac{\partial \mathbf{r}}{\partial u} du + \frac{\partial \mathbf{r}}{\partial v} dv) \cdot (\frac{\partial \mathbf{r}}{\partial u} du + \frac{\partial \mathbf{r}}{\partial v} dv)}\\ &= \sqrt{\frac{\partial \mathbf{r}}{\partial u} \cdot \frac{\partial \mathbf{r}}{\partial u} du^2 + 2 \frac{\partial \mathbf{r}}{\partial u} \cdot \frac{\partial \mathbf{r}}{\partial v} du dv + \frac{\partial \mathbf{r}}{\partial v} \cdot \frac{\partial \mathbf{r}}{\partial v} d^2} \end{aligned} \end{equation}\]
