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Examples

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

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

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

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



Vector Calculus
8/19/1998