Begin2.DVI

are defined and illustrated. Throughout the following discussion all surfaces are

considered to be oriented (two-sided) surfaces.

Consider a surface in space with an element of surface area dS constructed at

some general point on the surface as is illustrated in figure 7-16.

Figure 7-16. Element of surface area.

In the representation of various vector integrals, it is convenient to define vector

elements of surface area dS
whose magnitude is dS and whose direction is the same

as the unit outward normal ˆento the surface. Define this vector element of surface

area as dS
= ˆendS which can be considered as the limit associated with the area

∆
S=eˆn∆S.

Normal to a Surface

If ˆen is a normal to a smooth surface, then −ˆen is also normal to the surface.

That is, all smooth orientated surfaces possess two normals. If the surface is a

closed surface, there is an inside surface and an outside surface. The outside surface

is called the positive side of the surface. The unit normal to the positive side of a

surface is called the positive normal or outward normal. If the surface is not closed,

then one can arbitrarily select one side of the surface and call it the positive side,

therefore, the normal drawn to this positive side is also called the outward normal.