Definition:Vector Area

Definition
Let $S$ be a plane surface embedded in Cartesian $3$-space.

The vector area of $S$ is a technique of representing $S$ using a vector quantity $\mathbf S$.


 * Vector-area.png

The magnitude of the vector is determined by the area of $S$, while its direction is defined as the unit normal $\mathbf {\hat n}$ to the plane of $S$.

The sign of $\mathbf S = S \, \mathbf {\hat n}$ is determined by the right-hand rule according to the direction of rotation around $S$ when describing it.

The shape of $S$ is arbitrary.

While in the diagram above it is seen to be circular, it is usual to consider rectangles or whatever other shapes can be aligned conveniently with coordinate axes.