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
and
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
so that
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.