Put a paddlewheel into a moving body of water. Depending on the details of the flow, the paddlewheel might spin. Keeping its center fixed, change the orientation of the paddlewheel. There will be a preferred orientation, in which the paddlewheel spins the fastest.

We can investigate this situation for any vector field $\AA$. Pick a point $P$ and compute the circulation of $\AA$ around a small loop centered at $P$. Now change the orientation of the loop and do it again. The result depends on the size of the loop, so we divide by its area. There will be a preferred orientation in which the circulation per unit area will be a maximum. We define the curl of $\AA$, written $\grad\times\AA$, to be the vector whose direction is given by the normal vector to the plane in which the circulation is greatest, and whose magnitude is that circulation divided by the area of the loop, in the limit as the loop shrinks to a point. This construction yields the curl of $\AA$ at $P$, and we can repeat the process at any point; the curl of $\AA$ is a vector field.

What is the circulation per unit area at the point $P$ in an arbitrary direction? That's just the projection of $\grad\times\AA$ in the given direction. That is, $(\grad\times\AA)\cdot\Hat{u}$ is the (limiting value of) the circulation per unit area around a loop whose normal vector is $\Hat{u}$.

By choosing $\Hat{u}$ to be $\xhat$, $\yhat$, and $\zhat$ in turn, we can therefore compute the components of $\grad\times\AA$ in rectangular coordinates. For further details, see § {Curl}.