Active Calculus - Multivariable

In Preview Activity 12.8.1, you integrated the circulation density of a smooth vector field over a disk and found the result was equal to the circulation of the vector field along the region’s circular boundary. This relationship is the theme of this section. To see why this would make sense, consider the region \(R\) bounded by the curve \(C\) shown in Figure 12.8.1. We have placed a square grid inside \(R\) to suggest the idea of breaking \(R\) up into many smaller regions, most of which are square. This idea should make you think of the methods we have already seen of breaking up a region into smaller and smaller regions for Riemann sums.

The critical idea here is that if we integrate the circulation density over each of the small regions and add those up, this is the same as integrating the circulation density over the entirety of the region \(R\) because of the fundamental properties of integrals. Also, integrating circulation per unit area over a two-dimensional region should be related to the total circulation on that region. Now look at Figure 12.8.2 and think of these two square regions as being two of the square regions inside \(R\) in Figure 12.8.1.

Returning to Figure 12.8.1, if we find the circulation along the boundary of each of the smaller regions, the line integrals along the boundaries that lie inside the region \(R\) will all offset. Thus the value should equal \(\oint_C\vF\cdot d\vr\text{.}\) If we make our grid fine enough, all of the smaller regions into which \(R\) is divided will be very close to rectangular.

So far in this section, we have restricted ourselves to relatively nice closed curves when thinking about circulation. While the main theorem of this section will not allow us to consider arbitrary closed curves, it does cover more varied curves than we have discussed so far. A simple closed curve is a closed curve that does not cross itself, and these are the curves to which our next theorem applies.

At first glance, it may seem that Green’s Theorem is of purely intellectual interest. However, we have already encountered situations where parameterizing a curve can be complicated. This is particularly true when the curve has “corners” that require us to give separate parametrizations for several pieces of the curve, such as with the rectangular curve pictured in Figure 12.8.4. However, as with this rectangle, is often the case that a curve that is difficult to parametrize bounds a region that is not too complex to describe using rectangular or polar coordinates. For instance, the rectangular region in Figure 12.8.4 can be described as \(1\leq x\leq 4\) and \(2\leq y\leq 4\text{,}\) which is a simpler description than needing to parameterize each of the four sides of the rectangle separately. Additionally, the integrand of the double integral in Green’s Theorem involves partial derivatives that can sometimes result in an integrand that is easy to work with. The purpose of Green’s Theorem is, at its core, to allow you to exchange one type of integration problem (a line integral) for another type of integration problem (a double integral). This will be a recurring theme as this chapter continues.

Generally, Green’s Theorem is useful for allowing us to calculate line integrals by instead calculating a double integral. Going the other direction is harder, since finding a vector field \(\langle F_1,F_2\rangle\) so that \(\partial F_2/\partial x - \partial F_1/\partial y\) is equal to the integrand in the double integral can be difficult. However, the exercises will explore some situations where calculating a suitable line integral is a viable alternative to the double integral.

Notice that the assumptions in Green’s Theorem require that the region \(R\) be bounded by a simple closed curve \(C\) and that the vector field have continuous partial derivatives on \(R\) and \(C\text{.}\) In Exercise 7, we will explore what happens when the region \(R\) cannot be bounded by a simple closed curve, as sometimes multiple applications of Green’s Theorem can be used in those circumstances. Now, however, we will take a look at what happens in some cases where the vector field \(\vF\) is not smooth.

We can now see that Green’s Theorem is a powerful tool. However it cannot be used for all line integrals. In particular, the restriction that the vector field be smooth on the entire region bounded by a simple closed curve \(C\) creates limitations. This can occur even in cases where holes in the vector field’s domain or points where the vector field is not continuously differentiable lie away from \(C\text{.}\)

This section relies on the calculation of circulation around a closed curve from Section 12.2 and Section 12.3 as well as the idea of circulation density that was described in Section 12.6.