Physical Significance of Curl

In hydrodynamics, curl is sensed as rotation of a fluid and hence it is sometimes written as ‘rotation’ also. The curl of a vector field is sometimes called circulation or rotation (or simply not). If the fluid velocity vector v has a ‘curl’ somewhere, this means that the velocity field, over and above the general motion in a certain direction, has something link ↑^→_←↓ or ↑^→_←↓ superposed on it in that region, because the existence of ‘curl’ of the vector field at a point of space indicates the circulation or vorticity there.
The line integral of a conservative field A around any closed path is zero, i.e.,
            Φ A.dr = 0.

Therefore, the conservative vector fields have a zero curl at all points of space. That is why such field are known as non-curl field or lamellar vector fields.

An example of the conservative or lamellar vector field is electrostatic field E. An electrostatic field E can be expressed as gradient of a scalar field (Potential) V, i.e.,
            E = - ∇ V.

Now, curl E = - ∇ x E
        = ∇ x (∇E)
        = (∂/∂x i + ∂/∂y j + ∂/∂z k) x (∂V/∂x i + ∂V/∂y j + ∂V/∂z k)
        = |i j k ∂/∂x & ∂/∂ y& ∂/∂z @∂ V/∂x & ∂V/∂y & ∂V/∂z|

= i (∂2V/∂y∂z - ∂2V/∂z∂y) – j((∂2V/∂x∂z - ∂2V/∂z∂x) + k((∂2V/∂x∂y - ∂2V/∂y∂x) = 0.
Thus the curl of an electrostatic field is zero.
However, for electric fields generated by changing magnetic fields or for magnetic fields generated by electric currents, the curl is not zero.

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