Vector area

In 3-dimensional geometry, for a finite planar surface of scalar area and unit normal , the vector area is defined as the unit normal scaled by the area:

For an orientable surface S composed of a set of flat facet areas, the vector area of the surface is given by

where is the unit normal vector to the area .

For bounded, oriented curved surfaces that are sufficiently well-behaved, we can still define vector area. First, we split the surface into infinitesimal elements, each of which is effectively flat. For each infinitesimal element of area, we have an area vector, also infinitesimal.

where is the local unit vector perpendicular to . Integrating gives the vector area for the surface.

For a curved or faceted surface, the vector area is smaller in magnitude than the area. As an extreme example, a closed surface can possess arbitrarily large area, but its vector area is necessarily zero. Surfaces that share a boundary may have very different areas, but they must have the same vector area—the vector area is entirely determined by the boundary. These are consequences of Stokes theorem.