For bounded, oriented curved surfaces that are sufficiently well-behaved, we can schaum’s outline geometry pdf 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. 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. The concept of an area vector simplifies the equation for determining the flux through the surface. Consider a planar surface in a uniform field.
The flux can be written as the dot product of the field and area vector. This is much simpler than multiplying the field strength by the surface area and the cosine of the angle between the field and the surface normal.
Theory and problems of vector analysis. This page was last edited on 30 November 2016, at 23:54.