# Definition:Area Element

## Definition

An **area element** $\delta \mathbf S$ is a (usually) infinitesimal vector area,(usually) containing a particular point $P$.

In the cartesian $3$-space such an **area element** is modelled as a rectangle whose edges are parallel to two of the coordinate axes, and assigned lengths $\delta x$, $\delta y$ or $\delta z$ respectively.

### Outward Normal

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