10 - Surfaces

Smooth Parametrised Surface

Map $\mathbf{r}$ given by parametrisation

$$\mathbf{r}:U-{\mathbb{R}}^3:(u,v)\mapsto (x(u,v),y(u,v),z(u,v))$$

from an open subset $U\subseteq {\mathbb{R}}^2$ to ${\mathbb{R}}^3$ such that

• $x,y,z$ have continuous partial derivatives with respect to $u,v$ of all orders
• $\mathbf{r}$ is a bijection, with both $\mathbf{r}$ and ${\mathbf{r}}^{-1}$ being continuous
• the vectors

$$\frac{\partial \mathbf{r}}{\partial u}\text{ and }\frac{\partial \mathbf{r}}{\partial v}\text{ are linearly independent}$$

Tangent Plane

Let $\mathbf{r}:U\to {\mathbb{R}}^3$ be a smooth parametrised surface
Let $\mathbf{p}$ be a point on the surface

The plane containing $\mathbf{p}$ which is also parallel to vectors

$$\frac{\partial \mathbf{r}}{\partial u}(\mathbf{p})\text{ and }\frac{\partial \mathbf{r}}{\partial v}(\mathbf{p})$$

is known as the tangent plane to $\mathbf{r}(U)$ at $\mathbf{p}$

Note that it is well defined as the vectors are linearly independent

Normal Vectors to Surfaces

Any vector in the direction

$$\frac{\partial \mathbf{r}}{\partial u}(\mathbf{p})\wedge \frac{\partial \mathbf{r}}{\partial v}(\mathbf{p})$$

is normal to the surface at $\mathbf{p}$

Note that there are two unit normals of length $1$, just pointing in opposite directions

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Level Set

Let $f:{\mathbb{R}}^3\to \mathbb{R}$ be a function then

A level set is a set of points

$$\{(x,y,z)\in {\mathbb{R}}^3:f(x,y,z)=c\}$$

Where $c$ is a constant

Note that the level set is a surface in ${\mathbb{R}}^3$