Normal Vectors (Real Parametric Surfaces)#
Definition: Normal Vector
Let \(\phi: \mathcal{D} \subseteq \mathbb{R}^2 \to \mathbb{R}^3\) be a parametric surface which is totally differentiable at an interior point \(\boldsymbol{p}\) of \(\mathcal{D}\).
The normal vector of \(\phi\) at \(\boldsymbol{p}\) is the cross product of \(\phi\)'s partial derivatives there:
\[\partial_1 \phi(\boldsymbol{p}) \times \partial_2 \phi(\boldsymbol{p})\]
Theorem: Normal Vector and Regularity
Let \(\phi: \mathcal{D} \subseteq \mathbb{R}^2 \to \mathbb{R}^3\) be a parametric surface which is totally differentiable at an interior point \(\boldsymbol{p}\) of \(\mathcal{D}\).
Then \(\phi\) is regular at \(\boldsymbol{p}\) if and only if its normal vector there is non-zero:
\[partial_1 \phi(\boldsymbol{p}) \times \partial_2 \phi(\boldsymbol{p}) \ne \boldsymbol{0}\]
Proof
TODO