> [!NOTE] Lemma (Unit normal) > Let $S$ be a [[Parametrized Surface|surface parametrized]] by $\underline{r}(u,v).$ The unit normal at the point $P$ is given by $\hat{ \underline{n}} = \pm \frac{\underline{r}_{u} \times \underline{r}_{v}}{|\underline{r}_{u} \times \underline{r}_{v} |}$ Proof. At each point on the surface $r(u,v)$ where $(u,v)=(u_{0},v_{0}),$ the partial derivatives $\underline{r}_{u}(u_{0},v_{0})$ and $\underline{r}_{v}(u_{0},v_{0})$ are two distinct families of tangent vectors. The tangent plane $\underline{r}(u,v)$ spanned by these vectors. Thus $\underline{r}_{u}\times \underline{r}_{v},$ evaluated at $(u_{0},v_{0})$ is normal to the tangent plane.