In the previous module we saw how to compute the rate at which the total population inside a region was changing based on looking at what is happening inside the region but there is another way to make the same computation -- on the basis of the rate at which individuals are crossing the border of the region. As you might expect, these two methods yield the same result. The theorem that confirms this rather obvious fact is called the Divergence Theorem and was advertized in the prologue of this chapter. In this module we look at the Divergence Theorem in one, two, and three dimensions.
The Divergence Theorem in One Dimension
In one dimension we represent the population flow by a function P(x) which is positive at points where the population is migrating from left to right and negative at the points at which the population is migrating from right to left. The values of the function P(x) are measured in units like individuals per year. We can compute the rate at which the population is decreasing within an interval [a, b] by looking at what is happening at all the points inside the interval using the integral.
There is, however, a much easier way which relies on looking at only two points -- the two points on the border of the interval. The number P(a) represents the rate at which the population is entering the interval on the left and the number P(b) represents the rate at which the population is leaving the interval on the right. Thus, the quantity
represents the rate at which the population inside the region is decreasing. The Divergence Theorem says that these two ways of determining the rate at which the population inside the interval is decreasing yield the same result.
Divergence Theorem
This theorem is not very surprizing. In fact, it is obvious from our earlier remarks. This theorem is easily proved since it is just a restatement of the Fundamental Theorem of Calculus since
The Divergence Theorem in Two Dimensions
|
The movie at the right shows migration described by the vector field
We are interested in the migration into and out of the two triangular regions. The horizontal, and vertical sides of each region are 20 kilometers long and the two triangles are right triangles. Before reading on, determine the migration across each of the three sides of each triangular region. Then determine the net migration into (immigration) or out of (emigration) each of the two regions. After you have worked this problem and before reading on, read the answer by clicking the link below. |
|
Theorem
Suppose that migration is described by a constant vector field
and that a fence is represented by the vector L. Then the rate of migration across the fence is given by
Furthermore, suppose you look at this scene from the point of view of the observer in the figure above with the observer looking in the direction L. Then, if
is positive the migration is crossing the fence L from the observer's left to his or her right and if it is negative the migration is crossing the fence from right to left.
|
The movie at the right shows a somewhat more complicated situation in which migration is described by the vector field
and we are interested the migration into and out of a region like the circular blue region. In fact, we are interested in even more complicated situations in which the vector field is not constant -- that is, in which the rate and direction of migration may be different at different points. In this situation the simple wedge product is replaced by integration as you might expect. |
|
Theorem
Suppose that D is a region and C is its boundary. Suppose that C can be described by
and that as t goes from a to b, S(t) goes around the boundary counterclockwise with the inside of the region D on the left. Suppose that migration is described by the vector field
Then the rate of emigration out of the region D across its boundary C is given by
Proof
The proof can be found in the book Multivariable Calculus, Linear Algebra, and Differential Equations in a Real and Complex World or by clicking here for a pdf file.
We now have two different ways of determining the rate at which the population inside a region is decreasing.
The Diivergence Theorem in two dimensions states the unsurprizing fact that these two computations yield the same result.
Divergence Theorem
This theorem is sometimes called Green's Theorem. Actually it is just one-half of Green's theorem. The other half is discussed in the next module.
For each of the following problems migration is described by the indicated vector field F(x, y). Find the rate at which the total population in the indicated region D is decreasing two different ways, as described in the Divergance Theorem, by computing
and
Check that your two answers agree.
The Divergence Theorem in Three Dimensions
For the three dimensional version of the Divergence Theorem look in Section 5.3 of the book Multivariable Calculus, Linear Algebra, and Differential Equations in a Real and Complex World or at the same material in a pdf file by clicking here.
