Next: The scalar product
Up: Vectors
Previous: Vector algebra
Vector areas
Figure 4:

Suppose that we have planar surface of scalar area . We can define a vector
area whose magnitude is , and whose direction is perpendicular to
the plane, in the sense determined by the righthand grip rule on the rim (see Fig. 4).
This quantity clearly possesses both magnitude and direction. But is it a true
vector? We know that if the normal to the surface makes an angle with
the axis then the area seen looking along the direction is
.
This is the component of .
Similarly, if the normal makes an angle with the axis then the
area seen looking along the direction is
.
This is the component of . If we limit ourselves to a surface whose
normal is perpendicular to the direction then
.
It follows that
. If we rotate the
basis about the
axis by degrees, which is equivalent to rotating the normal to
the surface about the axis by degrees, then

(13) 
which is the correct transformation rule for the component of a vector. The
other components transform correctly as well. This proves
that a vector area is a
true vector.
According to the vector addition theorem, the projected area of two plane surfaces,
joined together at a line,
looking along the direction (say) is the component of the resultant of the vector areas of the two surfaces.
Likewise, for many joinedup plane areas, the projected area in the direction,
which is the same as the projected area of the rim in the direction, is the
component of the resultant of all the vector areas:

(14) 
If we approach a limit,
by letting the number of plane facets increase, and their areas reduce, then we
obtain a continuous surface denoted by the resultant vector area:

(15) 
It is
clear that the projected area of the rim in the direction is just .
Note that the rim of the surface determines the vector area rather than the nature of
the surface. So, two different surfaces sharing the same rim both possess the same
vector area.
In conclusion, a loop (not all in one plane) has a vector area which
is the resultant of the vector areas of any surface ending on the loop. The
components of are the projected areas of the loop in the
directions of the basis vectors. As a corollary, a closed surface has
,
since it does not possess a rim.
Next: The scalar product
Up: Vectors
Previous: Vector algebra
Richard Fitzpatrick
20060202