Definition:Area Element

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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.


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.