Tests for Conservative Vector Fields

The graphical test is not very accurate. It is almost impossible to tell if a three dimensional vector field is conservative in this fashion. We can use this idea to develop an analytical approach to testing whether a vector field is conservative or not. Recall that the curl is a way to measure a vector field's tendency to swirl. Thus, to eliminate the swirling, we want

$$\vec{\nabla} \times \vec{F} = \vec{0}$$ (2)

in order to ensure that $\vec{F}$ is a conservative vector field.

Let's check this condition for the vector fields in the example above.

$$\vec{F}(x,y) = \frac{x}{x^2 + y^2} \hat{i} + \frac{y}{x^2 + ... ... \qquad \qquad \vec{G}(x,y) = -y \hat{i} + x\hat{j} + 0\hat{k}.$$ (3)

$$\vec{\nabla} \times \vec{F} = \left\vert \begin{array} {ccc}... ...{2xy}{(x^2 y^2)^2} + \frac{2xy}{(x^2 + y^2)^2}\right) = \vec{0}$$ (4)

$$\vec{\nabla} \times \vec{G} = \left\vert \begin{array} {ccc}... ...y & x & 0 \end{array} \right\vert = \hat{k}(1 - -1) = 2 \hat{k}$$ (5)