Orientable Surfaces

# Orientable Surfaces

One type of surface which will be of significant importance later are known as orientable surfaces which we define below.

 Definition: An surface $\delta$ is said to be Orientable if there exists a unit normal field $\hat{N}(P)$ that is normal to ever point $P$ on $\delta$ as $P$ continuously varies over $\delta$.

For example, consider the following generic surface $S$ in $\mathbb{R}^3$: We can then construct a unit normal vector on $S$ which shows that $S$ is indeed orientable: Recall from the Surface Integrals page that if a surface $\delta$ is parameterized as $\vec{r}(u, v) = (x(u, v), y(u, v), z(u,v))$ then $\frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v}$ is normal to the surface $\delta$ for each point $P$ on $\delta$ and so we can define a natural unit normal field $\hat{N}$ on $\delta$ as:

(1)
\begin{align} \quad \hat{N} = \frac{\frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v}}{\biggr \| \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v} \biggr \|} \end{align}

For example, if a surface is given by the function $z = f(x, y)$, then as you should verify, the natural orientation of the surface generated by this function is given by:

(2)
\begin{align} \hat{N} = \frac{-\frac{\partial f}{\partial x} \vec{i} - \frac{\partial f}{\partial y} \vec{j} + \vec{k}}{\sqrt{\left ( \frac{\partial f}{\partial x} \right )^2 + \left ( \frac{\partial f}{\partial y} \right )^2 + 1}} \end{align}

The other orientation of the surface generated by $z = f(x, y)$ is given by $- \hat{N}$.

Since each surface has two "sides" per say and hence, two orientations. The choice of such orientation will matter when it comes to computing surface integrals of vector fields and physical interpretations of those integrals.

 Definition: If $\delta$ is an orientable surface with unit normal field $\hat{N} (P)$ then the Positive Side of $\delta$ is the side for which the unit normal vectors point outward/upward, and the Negative Side of $\delta$ is the side for which the unit normal vectors point inward/downward.

The following is a generic orientable surface showing both the positive and negative side of this surface. ## Induced Orientation of Boundary Curves

If $\delta$ is an orientable surface that has boundary curves, then the orientation of $\delta$ induces an orientation on the boundary curves. More specifically, if we're on the positive side of $\delta$ and we walk along any of its boundary curves, then the surface should always be to the left: 