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## The Laplacian

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
 (139)

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 steep-sided 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.