Vector Calculus: An Introduction

We saw in Section 8.2 that every conservative vector field is closed. We also mentioned that the converse is true if $D$ is simply connected. In this section we want to give a proof of this fact for plane vector fields. Recall from (8.5) that closed for a plane vector field, $\mathit{f}=(f_1,f_2)\text{,}$ means that

This is exactly the term appearing in (10.1). Simply connected means that every closed curve, $C\text{,}$ in $D$ can be continuously shrunk to a point in $D$ without ever leaving $D$ (see Figure 8.9). In particular this means that the domain, $D_C\text{,}$ enclosed by $C$ is a subset of $D\text{.}$ Hence it follows from (10.3) and Green’s Theorem that

This shows that if $D$ is simply connected and $\mathit{f}$ is closed then the integral over every closed piecewise continuous curve is zero. By Proposition 8.2 the field $\mathit{f}$ is conservative.