Exercise 2. Read the subsection "Assumptions on Curves, Vector Fields, and Domains" (p. 990). Explain what it means for a region to be "simply connected" in layman's terms.

990 Chapter 16 Integrals and Vector Fields property by indicating that the integral depends only on the initial and final points, and not on the path connecting them. Under reasonable differentiability conditions that we will specify, we will show that a field F is conservative if and only if it is the gradient field of a scalar function f-that is, if and only if F = Vf for some f. The function f then has a special name. DEFINITION If F is a vector field defined on D and F = V/ for some scalar function f on D, then f is called a potential function for F. A gravitational potential is a scalar function whose gradient field is a gravitational Simply connected .I field, an electric potential is a scalar function whose gradient field is an electric field, and so on. As we will see, once we have found a potential function f for a field F, we can evaluate all the line integrals in the domain of F over any path between A and B by (a) F . dr = / VS . de = [(B) - f(A). (1) If you think of VS for functions of several variables as analogous to the derivative S' for functions of a single variable, then you see that Equation (1) is the vector calculus rendition of the Fundamental Theorem of Calculus formula [ reds = fb) - fla). Simply connected Conservative fields have other important properties. For example, saying that F is conservative on D is equivalent to saying that the integral of F around every closed path in D is zero. Certain conditions on the curves, fields, and domains must be satisfied for Equa- (b) tion (1) to be valid. We discuss these conditions next.Assumptions on Curves, Vector Fields, and Domains In order for the computations and results we derive below to be valid, we must assume certain properties for the curves, surfaces, domains, and vector fields we consider. We give these assumptions in the statements of theorems, and they also apply to the examples and exercises unless otherwise stated. Not simply connected The curves we consider are piecewise smooth. Such curves are made up of finitely many smooth pieces connected end to end, as discussed in Section 13.1. For such curves we can compute lengths and, except at finitely many points where the smooth pieces con- nect, tangent vectors. We consider vector fields F whose components have continuous first (c) partial derivatives. The domains D we consider are connected. For an open region, this means that any two points in D can be joined by a smooth curve that lies in the region. Some results require D to be simply connected, which means that every loop in D can be contracted to a point in D without ever leaving D. The plane with a disk removed is a two-dimensional region that is not simply connected; a loop in the plane that goes around the disk cannot be contracted to a point without going into the "hole" left by the removed disk (see Figure 16.25c). Simi- larly, if we remove a line from space, the remaining region D is not simply connected. A Not simply connected curve encircling the line cannot be shrunk to a point while remaining inside D. Connectivity and simple connectivity are not the same, and neither property implies the other. Think of connected regions as being in "one piece" and simply connected (d) regions as not having any "loop catching holes." All of space itself is both connected and FIGURE 16.25 Four connected regions. simply connected. Figure 16.25 illustrates some of these properties. In (a) and (b), the regions are simply connected. In (c) and (d), the regions are Caution Some of the results in this chapter can fail to hold if applied to situations where not simply connected because the curves the conditions we've imposed do not hold. In particular, the component test for conserva- Ci and C cannot be contracted to a point tive fields, given later in this section, is not valid on domains that are not simply connected inside the regions containing them. (see Example 5). The condition will be stated when needed