6.3 Conservative vector fields  (Page 8/16)

When using [link] , it is important to remember that a theorem is a tool, and like any tool, it can be applied only under the right conditions. In the case of [link] , the theorem can be applied only if the domain of the vector field is simply connected.

To see what can go wrong when misapplying the theorem, consider the vector field from [link] :

This vector field satisfies the cross-partial property, since

and

Since F satisfies the cross-partial property, we might be tempted to conclude that F is conservative. However, F is not conservative. To see this, let

be a parameterization of the upper half of a unit circle oriented counterclockwise (denote this $C_1)$ and let

be a parameterization of the lower half of a unit circle oriented clockwise (denote this $C_2).$ Notice that $C_1$ and $C_2$ have the same starting point and endpoint. Since ${\text{sin}}^{2}t+{\text{cos}}^{2}t=1,$

and

Therefore,

Thus, $C_1$ and $C_2$ have the same starting point and endpoint, but ${\displaystyle {\int }_{C_1}\text{F}·d\text{r}}\ne {\displaystyle {\int }_{C_2}\text{F}·d\text{r}}.$ Therefore, F is not independent of path and F is not conservative.

To summarize: F satisfies the cross-partial property and yet F is not conservative. What went wrong? Does this contradict [link] ? The issue is that the domain of F is all of ${ℝ}^2$ except for the origin. In other words, the domain of F has a hole at the origin, and therefore the domain is not simply connected. Since the domain is not simply connected, [link] does not apply to F .

We close this section by looking at an example of the usefulness of the Fundamental Theorem for Line Integrals. Now that we can test whether a vector field is conservative, we can always decide whether the Fundamental Theorem for Line Integrals can be used to calculate a vector line integral. If we are asked to calculate an integral of the form ${\displaystyle {\int }_C\text{F}·d\text{r}},$ then our first question should be: Is F conservative? If the answer is yes, then we should find a potential function and use the Fundamental Theorem for Line Integrals to calculate the integral. If the answer is no, then the Fundamental Theorem for Line Integrals can’t help us and we have to use other methods, such as using [link] .