Orientation

Orientable Surface

A surface is orientable iff a continuous choice of unit normal vector can be made over the entire surface.

Two-sided surfaces are orientable. One-sided surfaces are non-orientable.

If $\sigma$ is smooth and orientable with unit normal $\hat{n}$ at each point, the direction of $\hat{n}$ can be chosen so that $\hat{n} = \hat{n}(x,y,z)$ varies continuously over the surface. Such a choice of unit normal field is an orientation of $\sigma$.

Orientation of a Smooth Parametric Surface

Let $\sigma$ be a smooth parametric surface given by $\boldsymbol{r}(u,v) = x(u,v)\hat{i} + y(u,v)\hat{j} + z(u,v)\hat{k}$. If

then

defines one orientation of $\sigma$. The opposite orientation is $-\hat{n}$.

Orientation of Non-Parametric Surfaces

Surfaces of the form $z = g(x,y)$, $y = g(x,z)$, $x = g(y,z)$ can be parametrized using their independent variables as parameters:

• $\boldsymbol{r}(x,y) = x\hat{i} + y\hat{j} + g(x,y)\hat{k}$, for $z = g(x,y)$
• $\boldsymbol{r}(x,z) = x\hat{i} + g(x,z)\hat{j} + z\hat{k}$, for $y = g(x,z)$
• $\boldsymbol{r}(y,z) = g(y,z)\hat{i} + y\hat{j} + z\hat{k}$, for $x = g(y,z)$

Each can be written as a level surface $G(x,y,z) = 0$ by moving all terms to one side. The vector

is a unit normal vector to the surface, whenever $\nabla G \ne \boldsymbol{0}$.