## Element of Surface Area in Terms of Intrinsic Coordinates

Suppose we have a surface defined in terms of intrinsic coordinates
$u,v$
,
$\mathbf{r} =\mathbf{r}(u,v)$
.
An element of surface area is
\begin{aligned} dS &= \left| \frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v} \right| du dv \\ &= \sqrt{(\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v}) \cdot (\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v})} du dv \end{aligned}

$(\mathbf{a} \times \mathbf{a}) \cdot (\mathbf{b} \times \mathbf{b}) = (\mathbf{a} \cdot \mathbf{a})(\mathbf{a} \cdot \mathbf{a}) - (\mathbf{a} \cdot \mathbf{b})(\mathbf{a} \cdot \mathbf{b})$

to obtain
\begin{aligned} dS &= \sqrt{(\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v}) \cdot (\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v})} du dv \\ &= \sqrt{(\frac{\partial \mathbf{r}}{\partial u} \cdot \frac{\partial \mathbf{r}}{\partial u}) (\frac{\partial \mathbf{r}}{\partial v} \cdot \frac{\partial \mathbf{r}}{\partial v}) -(\frac{\partial \mathbf{r}}{\partial u} \cdot \frac{\partial \mathbf{r}}{\partial v}) (\frac{\partial \mathbf{r}}{\partial u} \cdot \frac{\partial \mathbf{r}}{\partial v})} dudv \end{aligned}

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