The vector differential operator has the following form

We denote the curl of a vector field F this way

Notice here that both the differential operator and the field F are vector quantities. That means that the above is a cross product. Curl is therefore a vector.

If we have a field F which has the form

where i,j,k are unit vectors in the x,y and z directions, and where F_x,F_y and F_z are functions with partial derivatives, then the curl of F is given by

The above is the determinant form of the formula for curl. The first line is made up of unit vectors, the second of scalar operators, and the third of scalar functions, so this is not a determinant in the strict matematical sense.

Consider a vector field with only one component. We then get the situation shown in the applet below. This situation can happen for example in flowing water. Then the velocity of the water at different positions is the vector field. Try changing the gradient of the vector field, and see what the effect on the paddlewheel is.

You can see that the paddlewheel is responding to a local property of the vector field.

If you have an arbitrary vector field, then you can imagine the curl something like this:

The curl vector is perpendicular to the vector field.