Next: Curl
Up: Vectors
Previous: Divergence
So far we have encountered

(131) 
which is a vector field formed from a scalar field, and

(132) 
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
can write

(133) 
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

(134) 
where is the specific heat. It follows from the divergence theorem that

(135) 
Taking the divergence of both sides of Eq. (133), and making use of Eq. (135),
we obtain

(136) 
or

(137) 
If is constant then the above equation can be written

(138) 
The scalar field
takes the form
Here, the scalar differential operator

(140) 
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
operator).
What is the physical significance of the Laplacian? In one dimension,
reduces to
.
Now,
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
versa.
In two dimensions,

(141) 
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 steepsided valley in . Suppose that the bottom of the
valley runs parallel to the axis.
At the bottom of the
valley
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
value.
Let us now return to the heat conduction problem:

(142) 
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.,
. Thus,
the above heat conduction equation makes physical sense.
Next: Curl
Up: Vectors
Previous: Divergence
Richard Fitzpatrick
20060202