Exercise 3. Read the introduction to the subsection "Two Forms for Green's Theorem" (p. 1005 - 1006). According to the reading, what theorem is Green's theorem a 2D version of?

It's worth mentioning that while we mostly use Green's Theorem to convert a line integral into a double integral to make an easier computation, the reverse happens sometimes too. In particular, there is a formula used to compute the area of a region by using a line integral.2 See p. 1011 for details about this.

16.4 Green's Theorem in the Plane 1005 Finally, we let Ax and Ay approach zero to define the flux density of F at the point (x, y). div F is the symbol for divergence. The mathematical term for the flux density is the divergence of F. The symbol for it is div F, pronounced "divergence of F" or "div F." DEFINITION The divergence (flux density) of a vector field F = Mi + Nj at the point (x, y) is div F= AM aN ax + 3 ay (2) Source: div F(.);) 20 A gas is compressible, unlike a liquid, and the divergence of its velocity field mea- A gas capanding sures to what extent it is expanding or compressing at each point. Intuitively, if a gas is at the point (x. yol expanding at the point (x, y), the lines of flow would diverge there (hence the name) and, since the gas would be flowing out of a small rectangle about (x, ),), the divergence of F at (xp. )) would be positive. If the gas were compressing instead of expanding, the diver- gence would be negative (Figure 16.34). EXAMPLE 2 Find the divergence, and interpret what it means, for each vector field Sink: div F (1. y) 0, the gas is undergoing uniform expansion; if c