Oriented Surfaces

A surface is called orientable if a consistent choice of unit normal vector can be made at every point on the surface. A thin soap film stretched across a wire loop provides a familiar example. The film separates the air on one side from the air on the other, so one side can be selected as positive and the opposite side as negative. Once this ch...

A thin soap film stretches smoothly across a wire loop dipped into soapy water. The film separates the air on one side of the film from the air on the other. This means the surface is orientable.

At any point on this film, a unit normal vector can be assigned. Choosing one side for these arrows to point defines a positive orientation; choo...