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Concept
Vector area
Summary
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.
For a finite planar surface of scalar area S and unit normal n̂, the vector area S is defined as the unit normal scaled by the area:
For an orientable surface S composed of a set Si of flat facet areas, the vector area of the surface is given by
where n̂i is the unit normal vector to the area Si.
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 n̂ is the local unit vector perpendicular to dS. Integrating gives the vector area for the surface.
The vector area of a surface can be interpreted as the (signed) projected area or "shadow" of the surface in the plane in which it is greatest; its direction is given by that plane's normal.
For a curved or faceted (i.e. non-planar) surface, the vector area is smaller in magnitude than the actual surface 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.
The vector area of a parallelogram is given by the cross product of the two vectors that span it; it is twice the (vector) area of the triangle formed by the same vectors.
Official source
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