Section15.4Flow, Flux, Green’s Theorem and the Divergence Theorem
Subsection15.4.1Flow and Flux
Line integrals over vector fields have the natural interpretation of computing work when represents a force field. It is also common to use vector fields to represent velocities. In these cases, the line integral is said to represent flow.
Let the vector field represent the velocity of water as it moves across a smooth surface, depicted in Figure 15.4.1. A line integral over will compute “how much water is moving along the path .”
In the figure, “all” of the water above is moving along that curve, whereas “none” of the water above is moving along that curve (the curve and the flow of water are at right angles to each other). Because has nonzero horizontal and vertical components, “some” of the water above that curve is moving along the curve.
The “opposite” of flow is flux, a measure of “how much water is moving across the path .” If a curve represents a filter in flowing water, flux measures how much water will pass through the filter. Considering again Figure 15.4.1, we see that a screen along will not filter any water as no water passes across that curve. Because of the nature of this field, and each filter the same amount of water per second.
The terms “flow” and “flux” are used apart from velocity fields, too. Flow is measured by , which is the same as by Definition 15.3.2. That is, flow is a summation of the amount of that is tangent to the curve .
By contrast, flux is a summation of the amount of that is orthogonal to the direction of travel. To capture this orthogonal amount of , we use to measure flux, where is a unit vector orthogonal to the curve . (Later, we’ll measure flux across surfaces, too. For example, in physics it is useful to measure the amount of a magnetic field that passes through a surface.)
How is determined? We’ll later see that if is a closed curve, we’ll want to point to the outside of the curve (measuring how much is “going out”). We’ll also adopt the convention that closed curves should be traversed counterclockwise.
(If is a complicated closed curve, it can be difficult to determine what “counterclockwise” means. Consider Figure 15.4.3. Seeing the curve as a whole, we know which way “counterclockwise” is. If we zoom in on point , one might incorrectly choose to traverse the path in the wrong direction. So we offer this definition: a closed curve is being traversed counterclockwise if the outside is to the right of the path and the inside is to the left.)
Let be a vector field with continuous components defined on a smooth curve , parametrized by , let be the unit tangent vector of , and let be the clockwise 90degree rotation of .
Measuring flow is essentially the same as finding work performed by a force as done in the previous examples. Therefore we practice finding only flux in the following example.
Example15.4.5.Finding flux across curves in the plane.
Curves and each start at and end at , where follows the line and follows the unit circle, as shown in Figure 15.4.6. Find the flux across both curves for the vector fields and .
Figure15.4.6.Illustrating the curves and vector fields in Example 15.4.5. In (a) the vector field is , and in (b) the vector field is .
Solution.
We begin by finding parametrizations of and . As done in Example 15.3.12, parametrize by creating the line that starts at and moves in the direction: , for . We parametrize with the familiar on . For reference later, we give each function and its derivative below:
In Example 15.4.5, we saw that the flux across the two curves was the same when the vector field was . This is not a coincidence. We show why they are equal in Example 15.4.23. In short, the reason is this: the divergence of is 0, and when , the flux across any two paths with common beginning and ending points will be the same.
We also saw in the example that the flux across was 0 when the field was . Flux measures “how much” of the field crosses the path from left to right (following the conventions established before). Positive flux means most of the field is crossing from left to right; negative flux means most of the field is crossing from right to left; zero flux means the same amount crosses from each side. When we consider Figure 15.4.6.(b), it seems plausible that the same amount of was crossing from left to right as from right to left.
There is an important connection between the circulation around a closed region and the curl of the vector field inside of , as well as a connection between the flux across the boundary of and the divergence of the field inside . These connections are described by Green’s Theorem and the Divergence Theorem, respectively. We’ll explore each in turn.
Let be a closed, bounded region of the plane whose boundary is composed of finitely many smooth curves, let be a counterclockwise parametrization of , and let where and are continuous over . Then
Let and let be the region of the plane bounded by the triangle with vertices , and , shown in Figure 15.4.10. Verify Green’s Theorem; that is, find the circulation of around the boundary of and show that is equal to the double integral of over .
Figure15.4.10.The vector field and planar region used in Example 15.4.9
Solution.
The curve that bounds is composed of 3 lines. While we need to traverse the boundary of in a counterclockwise fashion, we may start anywhere we choose. We arbitrarily choose to start at , move to , etc., with each line parametrized by , and , respectively.
We leave it to the reader to confirm that the following parametrizations of the three lines are accurate:
,
for ,
with ,
,
for ,
with , and
,
for ,
with .
The circulation around is found by summing the flow along each of the sides of the triangle. We again leave it to the reader to confirm the following computations:
and .
The circulation is the sum of the flows: .
We confirm Green’s Theorem by computing . We find . The region is bounded by the lines , and . Integrating with the order is most straightforward, leading to
,
which matches our previous measurement of circulation.
Figure15.4.13.The vector field and planar region used in Example 15.4.12
Solution.
Computing the circulation directly using the line integral looks difficult, as the integrand will include terms like “.”
Green’s Theorem states that ; since in this example, the double integral is simply 0 and hence the circulation is 0.
Since , we can conclude that the circulation is 0 in two ways. One method is to employ Green’s Theorem as done above. The second way is to recognize that is a conservative field, hence there is a function wherein . Let be any point on the curve ; since is closed, we can say that “begins” and “ends” at . By the Fundamental Theorem of Line Integrals, .
One can use Green’s Theorem to find the area of an enclosed region by integrating along its boundary. Let be a closed curve, enclosing the region , parametrized by . We know the area of is computed by the double integral , where the integrand is . By creating a field where , we can employ Green’s Theorem to compute the area of as .
Another interesting scenario that comes up is the case of multiply-connected regions (as opposed to simply-connected). If a bounded region has a “hole”, its boundary will consist of more than one curve: the outer boundary, as well as the boundary of the hole. Green’s Theorem applies in this situation as well, as the video in Figure 15.4.19 explains.
Green’s Theorem makes a connection between the circulation around a closed region and the sum of the curls over . The Divergence Theorem makes a somewhat “opposite” connection: the total flux across the boundary of is equal to the sum of the divergences over .
Theorem15.4.20.The Divergence Theorem (in the plane).
Let be a closed, bounded region of the plane whose boundary is composed of finitely many smooth curves, let be a counterclockwise parametrization of , and let where and are continuous over . Then
Let , let be the circle of radius 2 centered at the origin and define to be the interior of that circle, as shown in Figure 15.4.22. Verify the Divergence Theorem; that is, find the flux across and show it is equal to the double integral of over .
Let be any field where , and let and be any two nonintersecting paths, except that each begin at point and end at point (see Figure 15.4.24). Show why the flux across and is the same.
By referencing Figure 15.4.24, we see we can make a closed path that combines with , where is traversed with its opposite orientation. We label the enclosed region . Since , the Divergence Theorem states that
.
Using the properties and notation given in Theorem 15.3.10, consider:
Figure15.4.24.As used in Example 15.4.23, the vector field has a divergence of 0 and the two paths only intersect at their initial and terminal points.
(where is the path traversed with opposite orientation)
In this section, we have investigated flow and flux, quantities that measure interactions between a vector field and a planar curve. We can also measure flow along spatial curves, though as mentioned before, it does not make sense to measure flux across spatial curves.
It does, however, make sense to measure the amount of a vector field that passes across a surface in space — i.e, the flux across a surface. We will study this, though in the next section we first learn about a more powerful way to describe surfaces than using functions of the form .
In the following exercises, a vector field and a closed curve , enclosing a region , are given. Verify Green’s Theorem by evaluating and , showing they are equal.
In the following exercises, a vector field and a closed curve , enclosing a region , are given. Verify the Divergence Theorem by evaluating and , showing they are equal.