Physical Interpretation of the Curl

The curl of a vector field measures the tendency for the vector field to swirl around. Imagine that the vector field represents the velocity vectors of water in a lake. If the vector field swirls around, then when we stick a paddle wheel into the water, it will tend to spin. The amount of the spin will depend on how we orient the paddle. Thus, we should expect the curl to be vector valued.

As other examples, consider the illustrations below. The field on the left, called $\vec{F}$ has curl with positive $\hat{k}$ component. To see this, use the right hand rule. Place your right hand at P. Point your fingers toward the tail of one of the vectors of $\vec{F}$. Now curl your fingers around in the direction of the tip of the vector. Stick your thumb out. It points toward the +z axis, so the curl should have positive $\hat{k}$component.

The second vector field has no swirling tendency at all (from visual inspection) so we would expect $\nabla \times \vec{G} = \vec{0}$. The third vector field doesn't look like it swirls either, so it also has zero curl.