Curl

Happy childhood years... a merry-go-round, a child... two grandmothers and two grandfathers spin the merry-go-round while the happy little one rides on it. This is how we will begin our explanation of the vector operator “curl”. Let us make the picture a bit more concrete. A merry-go-round is an object that rotates about its own axis and accordingly exists in a single plane formed by the coordinate system (x,y)(x, y). The merry-go-round is a disk whose tangents are parallel to the axes and pass through the points (−Δx,0),(Δx,0),(0,−Δy),(0,Δy)(-\Delta x, 0), (\Delta x, 0), (0, -\Delta y), (0, \Delta y), while the center of the merry-go-round is located at the center of the coordinate system. At these four points stand the grandmothers and grandfathers, and they push the merry-go-round.

It is intuitively clear that if neither grandfather nor either grandmother spins it, the merry-go-round will not rotate, but it is also clear that if all four spin it not in the same direction but, say, the grandfathers counterclockwise and the grandmothers clockwise, then, provided their forces are equal, the merry-go-round again will not rotate... and the little one will probably cry. Hmm...

The angular velocity imparted to the merry-go-round equals the ratio of the difference of the linear velocities to the distance from the center to the point of application of these velocities

V(Δx,0)−V(−Δx,0)Δx\displaystyle \dfrac{V_{(\Delta x, 0)} - V_{(-\Delta x, 0)}}{\Delta x} −V(0,Δy)−V(0,−Δy)Δy\displaystyle - \dfrac{V_{(0, \Delta y)} - V_{(0, -\Delta y)}}{\Delta y}
(1)

Now let us send the radius of the merry-go-round to zero and pass from finite differences to derivatives. Thus, the magnitude of the merry-go-round's angular velocity vector will be determined by the value

ωz\displaystyle \omega_z =∂Vy∂x\displaystyle = \dfrac{\partial V_y}{\partial x} −∂Vx∂y\displaystyle - \dfrac{\partial V_x}{\partial y}
(2)

Recalling the right-hand rule, we obtain the following triple of vectors (x,y,z),(y,z,x),(z,x,y)(x,y,z), (y,z,x), (z,x,y). Let us rewrite the above formula in vector form

(ωx,ωy,ωz)\displaystyle (\omega_x, \omega_y, \omega_z) =(∂Vz∂y−∂Vy∂z,∂Vx∂z−∂Vz∂x,∂Vy∂x−∂Vx∂y)\displaystyle = \left(\dfrac{\partial V_z}{\partial y} - \dfrac{\partial V_y}{\partial z}, \dfrac{\partial V_x}{\partial z} - \dfrac{\partial V_z}{\partial x}, \dfrac{\partial V_y}{\partial x} - \dfrac{\partial V_x}{\partial y}\right)
(3)

The resulting vector of angular velocities is called the curl

rot F\displaystyle rot \space F =(∂Fz∂y−∂Fy∂z,∂Fx∂z−∂Fz∂x,∂Fy∂x−∂Fx∂y)\displaystyle = \left(\dfrac{\partial F_z}{\partial y} - \dfrac{\partial F_y}{\partial z}, \dfrac{\partial F_x}{\partial z} - \dfrac{\partial F_z}{\partial x}, \dfrac{\partial F_y}{\partial x} - \dfrac{\partial F_x}{\partial y}\right)
(4)

The curl shows how much the field swirls at a specific point; for example, a field swirls around a conductor carrying a current. Generally speaking, the curl can be represented as the cross product of the nabla operator with the vector

rot F\displaystyle rot \space F =∇×F\displaystyle = \nabla × F
(5)