integral for the vector area can be derived: where is the boundary of , i.e. one or more oriented closed space curves. This is analogous to the two dimensional area calculation using Green's theorem. 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 : For example, the projected area onto the -plane is equivalent to the -component of the vector area, and is also equal to where is the angle between the plane normal and the -axis. 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 Area Vectors (mathematics and physics) Analytic geometry