Definition:Vector Area
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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 $\mathbf S$.
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.
Also see
- Results about vector area can be found here.
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {II}$: The Products of Vectors: $5$. Vector Area