Definition:Area Element/Outward Normal
Definition
Let $\delta \mathbf S$ be an area element embedded in a coordinate frame with position vector $\mathbf r$.
The outward normal of $\delta \mathbf S$ is defined to be the normal vector $\mathbf n$ to $\delta \mathbf S$ such that:
- $\mathbf r \cdot \mathbf n > 0$
where $\cdot$ denotes the dot product.
In the event that $\mathbf r \cdot \mathbf n = 0$, the outward normal may be chosen arbitrarily.
Examples
Surface of Body
An area element $\delta \mathbf S$ is often coincident or approximately coincident with part of the surface of a body in space.
Such a body can be considered to have the whole of $S$ covered by such as $\delta \mathbf S$.
The direction of the normal to such a $\delta \mathbf S$ is conventionally taken to be the outward normal of $B$.
If $B$ is complicated in shape, a redefinition of the term outward normal may be appropriate.
Also see
- Results about area elements can be found here.
Sources
- 1990: I.S. Grant and W.R. Phillips: Electromagnetism (2nd ed.) ... (previous) ... (next): Chapter $1$: Force and energy in electrostatics: $1.4$ Gauss's Law: $1.4.2$ The flux of the electric field out of a closed surface