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vector area : ウィキペディア英語版
vector area
In 3-dimensional geometry, for a finite planar surface of scalar area S and unit normal \hat, the vector area \mathbf is defined as the unit normal scaled by the area:
:\mathbf = \mathbf = \sum_i \mathbf}_i is the unit normal vector to the area S_i.
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.
:d\mathbf = \mathbf} is the local unit vector perpendicular to dS. Integrating gives the vector area for the surface.
:\mathbf = \int d\mathbf
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.〔Murray R. Spiegel, ''Theory and problems of vector analysis,'' Schaum's Outline Series, McGraw Hill, 1959, p. 25.〕 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.
== Projection of area onto planes ==
The projected area onto (for example) the ''x''-''y'' plane is equivalent to the ''z''-component of the vector area, and is given by
:\mathbf = \left| \mathbf \right| \cos \theta
where \theta is the angle between the plane normal and the ''z''-axis.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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