A Look at Conservative Vector Fields

Graphically, a vector field is conservative if it has no tendency to ``swirl around.'' If it did swirl, then the value of the line integral would be path dependent. A conservative vector field has the direction of its vectors more or less evenly distributed. For example, let

$$\vec{F}(x,y) = \frac{x}{x^2 + y^2} \hat{i} + \frac{y}{x^2 + y^2} \hat{j} \qquad \qquad \vec{G}(x,y) = -y \hat{i} + x\hat{j}.$$ (1)

The graphs of these vector fields are shown below. It is easy to see that $\vec{F}$ is a radial vector field, and thus has no tendency to swirl. On the other hand, $\vec{G}$ definitely swirls around. Note that if we compute $\int_{C_1} \vec{G}\cdot d\vec{r}$ we get a positive value since we are traveling along the path in the direction of the vector field. If we compute the line integral along C₂ however, we get a negative. We have two paths between identical endpoints with different values of the line integral, so $\vec{G}$ cannot be independent of path. Thus, $\vec{G}$ is not conservative.