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Tangent Planes (Real Parametric Surfaces)#

Definition: Tangent Plane

Let \(\phi: \mathcal{D} \subseteq \mathbb{R}^2 \to \mathbb{R}^n\) be a parametric surface which is regular at an interior point \(\boldsymbol{p}\) of \(\mathcal{D}\).

The tangent plane of \(\phi\) at \(\boldsymbol{p}\) is the span of its partial derivatives there:

\[\operatorname{span}\{\partial_1 \phi(\boldsymbol{p}), \partial_2 \phi(\boldsymbol{p}) \}\]