vector area

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.

For bounded, oriented curved surfaces that are sufficiently well-behaved, we can still define vector area.

(Vector area is also denoted by A rather than S, but this conflicts with the magnetic potential, a separate vector field).

S and S are the vector areas for the surfaces S and S respectively.

The vector area is a combination of the magnitude of the area through which the mass passes through, A, and a unit vector normal to the area, .

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.

In this definition, flux is generally a vector due to the widespread and useful definition of vector area, although there are some cases where only the magnitude is important (like in number fluxes, see below).

Suppose a planar closed loop carries an electric current and has vector area (, , and coordinates of this vector are the areas of projections of the loop onto the , , and planes).

These ambiguities are resolved by the right-hand rule: With the palm of the right-hand toward the area of integration, and the index-finger pointing along the direction of line-integration, the outstretched thumb points in the direction that must be chosen for the vector area dS.