## Constructing a Normal to a Surface From Partial Derivatives

For a surface
$\mathbf{r}=\mathbf(\alpha , \beta )$
, a normal vector is
$\mathbf{N} =\frac{\partial \mathbf{r}}{\partial \alpha} \times \frac{\partial \mathbf{r}}{\partial \beta}$
.
To see this, at a point
$(\alpha_0 , \beta_0 )$
in the surface,
$\frac{\partial \mathbf{r}}{\partial \alpha}_{(\alpha_0 , \beta_0 )}$
and
$\frac{\partial \mathbf{r}}{\partial \beta}_{(\alpha_0 , \beta_0 )}$
are both tangential to the surface so
$\mathbf{N} =\frac{\partial \mathbf{r}}{\partial \alpha}|_{(\alpha_0 , \beta_0 )} \times \frac{\partial \mathbf{r}}{\partial \beta}|_{(\alpha_0 , \beta_0 )}$
is normal to the surface.