If I have u = ( u 1 , u 2 ) is a vector...
If I have is a vector of two components, then what is ? where
I think it should give a scalar and not a vector.
Answer & Explanation
The curl of a vector field is only really defined on vector fields . For a two-dimensional field the z component is taken to be zero. The resulting curl has non-zero components only in the z-direction:
so people sometimes call the following scalar quantity the "curl":
The curl of a 2D vector can be defined with the following "intuition": The 2D vector can be "extended" to as . Now we can take the "usual" curl of this vector: and we identify with the curl of u, i.e.