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Oriented Surfaces

A surface is said to be oriented (when this is possible) if a direction of positive flow has been chosen. To choose a direction of positive flow we specify a normal vector to the surface. Any flow that is in the general direction of the normal vector is considered positive and any flow which heads against the normal vector is considered negative. Note not all surfaces are orientable (e.g. the Möbius band)

Examples:

1.

The plane z=0 (the xy plane) has two possible orientations, up or down. Every normal vector looks like $\vec{n}=a\hat{k}$. If a is positive then the normal points up and if a is negative then the normal points down. As an example consider the upward normal $\vec{n}=\hat{k}$. We can simplify our lives a little by always using unit normal vectors. If we do this then there are only two choices for the normal vector, up or down.

2.

Consider the sphere x²+y²+z²=1. For closed surfaces we usually take the normal vector to point outwards. In this case a vector that points radially outward will be a normal vector.

$$\hat{n} =\frac{\vec{r}}{\left\Vert \vec{r}\right\Vert }$$

$$=\frac{x}{\sqrt{x^{2}+y^{2}+z^{2}}}\hat{\imath}+\frac{y}{\sq... ...+y^{2}+z^{2}}}\hat{j}+\frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}}\hat{k}$$

and any flow out of the sphere is considered positive while flow into the sphere is considered negative.