Examples

Example 1. Compute the curl of $\vec{F} = -y \hat{i} + x \hat{j}$.

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

This is the vector field on the left in the figure above. As you can see, the analytical approach does bear out that the curl has positive $\hat{k}$ component.

Example 2. Compute the curl of $\vec{H}(\vec{r}) = \vec{r}$. Now, $\vec{H} = x\hat{i} + y\hat{j} + z\hat{k}$ so

$$\nabla \times \vec{H} = \left\vert \begin{array} {ccc}\hat{i... ...tial}{\partial z}\ x & y & z \end{array} \right\vert = \vec{0}$$ (6)

This is, as you have no doubt guessed, the vector field on the far right in the above illustration.