Vector area
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In
Every
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 S and
For an
areas, the vector area of the surface is given byFor bounded, oriented curved surfaces that are sufficiently
Properties
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
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. In general, the vector area of any surface whose boundary consists of a sequence of straight line segments (analogous to a polygon in two dimensions) can be calculated using a series of cross products corresponding to a triangularization of the surface. This is the generalization of the Shoelace formula to three dimensions.
Using Stokes' theorem applied to an appropriately chosen vector field, a boundary integral for the vector area can be derived:
Applications
Area vectors are used when calculating surface integrals, such as when determining the flux of a vector field through a surface. The flux is given by the integral of the dot product of the field and the (infinitesimal) area vector. When the field is constant over the surface the integral simplifies to the dot product of the field and the vector area of the surface.
Projection of area onto planes
The projected area onto a plane is given by the dot product of the vector area S and the target plane unit normal m̂:
See also
- Bivector, representing an oriented area in any number of dimensions
- De Gua's theorem, on the decomposition of vector area into orthogonal components
- Cross product
- Surface normal
- Surface integral
Notes
- ^ Spiegel, Murray R. (1959). Theory and problems of vector analysis. Schaum's Outline Series. McGraw Hill. p. 25.