Tangent Space#

Definition: Tangent Space

Let \(S: \mathcal{D} \subseteq \mathbb{R}^2 \to \mathbb{R}^n\) be a parametric surface which is differentiable at \(\mathbf{a} \in \mathcal{D}\).

The tangent space of \(S\) at \(\mathbf{a}\) is the span of its partial derivatives at \(\mathbf{a}\) with respect to Cartesian coordinates:

\[ \mathop{\operatorname{span}}\left(\frac{\partial S}{\partial x}(\mathbf{a}), \frac{\partial S}{\partial y}(\mathbf{a})\right) \]

Notation

\[ T_S(\mathbf{a}) \qquad T_{S(\mathbf{a})}S \]

Theorem: Dimension of the Tangent Space

Let \(S: \mathcal{D} \subseteq \mathbb{R}^2 \to \mathbb{R}^n\) be a parametric surface which is differentiable at \(\mathbf{a} \in \mathcal{D}\).

The tangent space of \(S\) at \(\mathbf{a}\) is at most \(2\)-dimensional.

Proof

TODO

Normal Spaces#

Definition: Normal Space

Let \(S: \mathcal{D} \subseteq \mathbb{R}^2 \to \mathbb{R}^n\) be a parametric surface which is differentiable at \(\mathbf{a} \in \mathcal{D}\).

The normal space of \(S\) at \(\mathbf{a}\) is the orthogonal complement of \(S\)'s tangent plane at \(\mathbf{a}\).

\[ N_S(\mathbf{a}) \qquad N_{S(\mathbf{a})}S \]

Definition: Surface Normals

The elements of the normal space \(N_{S(\mathbf{a})}S\) are known as the surface normals or normal vectors of \(S\) at \(\mathbf{a}\).

Theorem: Surface Normals in 3D

Let \(S: \mathcal{D} \subseteq \mathbb{R}^2 \to \mathbb{R}^n\) be a parametric surface which is differentiable at \(\mathbf{a} \in \mathcal{D}\).

If \(n = 3\), then the normal space of \(S\) at \(\mathbf{a}\) is spanned by the cross product of \(S\)'s partial derivatives at \(\mathbf{a}\) with respect to Cartesian coordinates:

\[ N_S(\mathbf{a}) = \mathop{\operatorname{span}}\left\{\frac{\partial S}{\partial x}(\mathbf{a})\times\frac{\partial S}{\partial y}(\mathbf{a})\right\} \]

Note

When dealing with parametric surfaces in 3D, it is very common to use the terms "surface normal" and "normal vector" for this cross product.

Notation

In such cases, we denote this cross product by \(\mathbf{N}(\mathbf{a})\) and its normalization by \(\mathbf{n}(\mathbf{a})\)

Proof

TODO

Definition: Surface Area

Let \(s: \mathcal{D} \subseteq \mathbb{R}^2 \to \mathbb{R}^3\) be a parametric surfaces.

If \(s\) is differentiable on \(\mathcal{D}\), then its surface area is the double integral of the magnitude of its normal vector:

\[ \iint_{\mathcal{D}} ||\mathbf{N}|| \]