So far we have encountered
which is a vector field formed from a scalar field, and
which is a scalar field formed from a vector field. There are two ways in which
we can combine and div. We can either form the vector field
or the scalar field
The former is not particularly interesting, but the scalar field
turns up in a great many physics problems, and is,
therefore, worthy of discussion.
Let us introduce the heat flow vector , which is the rate of flow of heat
energy per unit area across a surface perpendicular to the direction of .
In many substances, heat flows directly down the temperature gradient, so that we
where is the thermal conductivity. The net rate of heat flow
out of some closed surface must be equal
to the rate of decrease of heat energy in the volume enclosed by .
Thus, we can write
where is the specific heat. It follows from the divergence theorem that
Taking the divergence of both sides of Eq. (133), and making use of Eq. (135),
If is constant then the above equation can be written
The scalar field
takes the form
Here, the scalar differential operator
is called the Laplacian. The Laplacian is a good scalar operator
(i.e., it is coordinate independent) because it is formed from a
combination of div
(another good scalar operator) and (a good vector
What is the physical significance of the Laplacian? In one dimension,
is positive if is
concave (from above) and negative if it is convex. So, if is less than the
average of in its surroundings then is positive, and vice
In two dimensions,
Consider a local minimum of the temperature.
At the minimum, the slope of increases in all directions, so is
positive. Likewise, is negative at
a local maximum.
Consider, now, a steep-sided valley in . Suppose that the bottom of the
valley runs parallel to the -axis.
At the bottom of the
is large and positive, whereas
is small and may even be negative. Thus,
is positive, and this is associated with being less than the average local
Let us now return to the heat conduction problem:
It is clear that if is positive then is locally less than the
average value, so
: i.e., the region heats up.
Likewise, if is negative then is locally greater than the average
value, and heat flows out of the region: i.e.,
the above heat conduction equation makes physical sense.