Vector Calculus: An Introduction

Checking path independence for a given vector field by evaluating the integral over every closed curve is an impossible task! Hence, in practice, we need other criteria to check whether a given vector field is conservative. In this section we want to introduce such a criterion.

Hence for a vector field $\mathit{f}$ to be conservative, (8.4) must be satisfied. The question arising is whether (8.4) guarantees that $\mathit{f}$ is conservative. Unfortunately, the answer is NO in general (see Example 8.12 below)! However, under additional assumptions on $D$ we will be able to show that (8.4) is sufficient to ensure that $\mathit{f}$ is a gradient field. Because of their significance in any `field theory,’ vector fields satisfying condition (8.4) have a special name.

We mentioned above that it depends on the domain $D$ whether every closed vector field on $D$ is conservative or not. The condition on $D$ is a topological one. We need to make sure that every closed curve in $D$ can be continuously shrunk to a point in $D$ without ever leaving $D\text{.}$ Figure 8.9 shows a simply connected domain in the plane. If a plane domain has a hole like the one in Figure 8.10, a curve around the hole cannot be shrunk to a point in $D\text{.}$

One can then prove the following theorem.

A proof of the above theorem in full generality is out of the scope of these notes. Some special cases can easily be obtained as applications of the theorems of Green and Stokes in Part IV. For $N=2$ and $N=3$ we will give a complete proof in Section 10.4 and Section 13.3.In the following sections we look at the special cases of $N=2$ and $N=3\text{.}$