If I have u = ( u 1 , u 2 ) is a vector...

owsicag7

owsicag7

Answered

2022-07-18

If I have u = ( u 1 , u 2 ) is a vector of two components, then what is curl u := × u? where = ( x , y )
I think it should give a scalar and not a vector.

Answer & Explanation

abortargy

abortargy

Expert

2022-07-19Added 19 answers

The curl of a vector field is only really defined on vector fields F : R 3 R 3 . For a two-dimensional field u = ( u 1 ( x , y ) , u 2 ( x , y ) )the z component is taken to be zero. The resulting curl has non-zero components only in the z-direction:
× ( u 1 , u 2 , 0 ) = ( 0 , 0 , u 2 x u 1 y )
so people sometimes call the following scalar quantity the "curl":
u 2 x u 1 y
PoentWeptgj

PoentWeptgj

Expert

2022-07-20Added 6 answers

The curl of a 2D vector can be defined with the following "intuition": The 2D vector u = ( u 1 , u 2 ) can be "extended" to R 3 as u ^ = ( u 1 , u 2 , 0 ). Now we can take the "usual" curl of this u ^ vector: v ^ := × u ^ = ( 0 , 0 , v 3 ) and we identify v 3 with the curl of u, i.e.
curl ( u 1 , u 2 ) = ε 321 2 u 1 + ε 312 1 u 2 = u 2 x u 1 y

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