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.
where n̂ is the local unit vector perpendicular to dS.
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.
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.
Using Stokes' theorem applied to an appropriately chosen vector field, a boundary integral for the vector area can be derived:
is the boundary of S, i.e. one or more oriented closed space curves.
This is analogous to the two dimensional area calculation using Green's theorem.
The flux is given by the integral of the dot product of the field and the (infinitesimal) area vector.
The projected area onto a plane is given by the dot product of the vector area S and the target plane unit normal m̂:
where θ is the angle between the plane normal n̂ and the z-axis.