Vector area
In 3-dimensional geometry and vector calculus, an area vector is a vector combining an area quantity with a direction, thus representing an oriented area in three dimensions. Every bounded surface in three dimensions can be associated with a unique area vector called its vector area. It is equal to the surface integral of the surface normal, and distinct from the usual (scalar) surface area. Vector area can be seen as the three dimensional generalization of signed area in two dimensions. Definition 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 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. Properties The vector area