Vector Calculus: An Introduction

In Section 8.2 we stated without a proof that a closed vector field, $\mathit{f}\text{,}$ on a simply connected domain $D\subset {\mathbb{R}}^3$ is conservative. We now want to apply the theorem of Stokes to sketch a proof. Simply connected means that every closed curve in $D$ can be continuously deformed into a point in $D$ without ever leaving $D$ as shown in Figure 8.9. It turns out that this is equivalent to the following. Suppose that $C_0$ and $C_1$ are smooth curves in $D$ connecting ${\mathit{x}}_0\in D$ to ${\mathit{x}}_1\in D\text{.}$ Then it is possible to deform $C_0$ smoothly into $C_1$ without leaving $D\text{.}$ When doing this the “trace” left by the moving curve defines a surface, $S\text{,}$ with boundary $\partial S=C_0\cup C_1$ as shown in Figure 13.5.