Curl and its physical significance

Definition: The curl of a vector field A is a vector whose magnitude is the maximum net circulation of A per unit area, as the area shrinks to zero around a point, and whose direction is the normal to that area (right-hand rule).

(∇ × A) · a_n = lim (ΔS → 0)  [ ∮ A · dl ] / ΔS     (path in the plane for which this is maximum)

In Cartesian coordinates:

         | a_x    a_y    a_z  |
∇ × A =  | ∂/∂x   ∂/∂y   ∂/∂z |
         | A_x    A_y    A_z  |

      = (∂A_z/∂y − ∂A_y/∂z) a_x + (∂A_x/∂z − ∂A_z/∂x) a_y + (∂A_y/∂x − ∂A_x/∂y) a_z

Physical significance

• Curl measures the rotation (swirl, whirl) of the field about a point. A non-zero curl means the field lines tend to form closed loops around that point.
• The magnitude tells how strong the rotation is, and the direction is the axis of rotation (right-hand rule).
• If ∇ × A = 0, the field is irrotational (conservative): the line integral between two points is path independent, and A can be written as the gradient of a scalar, A = −∇V. Static electric field E is of this type (∇ × E = 0).
• If ∇ × A ≠ 0, the field is rotational. Magnetic field H around a current is of this type (∇ × H = J).

Example

Consider water in a river or a whirlpool, with velocity v. Place a tiny paddle wheel in the water.

   uniform flow            whirlpool
   → → → →                 ↗ → ↘
   → → → →                 ↑ ● ↓      paddle wheel at ●
   → → → →                 ↖ ← ↙      rotates: curl ≠ 0
   wheel does not rotate: curl = 0

• In the uniform flow the wheel only moves along; it does not spin, so ∇ × v = 0.
• In a whirlpool the wheel spins. The speed of spin is proportional to |∇ × v|, and the axis of the wheel gives the direction of curl.

Another example: for A = −y a_x + x a_y, ∇ × A = (∂(x)/∂x − ∂(−y)/∂y) a_z = 2 a_z. The field circulates counter-clockwise about the z-axis, and the curl points along +z.