Curl in 2D is a vector or a scalar?

In summary, the equation for the curl of a vector field in 2D is \oint_{s}\vec{f}\cdot d\vec{s} = \iint_{A}\vec{\nabla}\times \vec{f} d^2A, where the difference between ##|\vec{\nabla}\times \vec{f}|## and ##\vec{\nabla}\times \vec{f}## is that ##\vec{\nabla}\times \vec{f}## is the modulus.
  • #1
Jhenrique
685
4
Vector, by definition, have 2 or 3 scalar components (generally), but the curl of a vector field f(x,y) in 2D have only one scalar component: [tex]\left ( \frac{\partial f_y}{\partial x} -\frac{\partial f_x}{\partial y} \right )dxdy[/tex]
So, the Curl of a vector field in 2D is a vector or a scalar?
 
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  • #2
You can think of 2D curl as a special case of 3D where the only component is in the k direction.

In Green's theorem, for example, the normal to the area is also k, so the integral reduces to the scalar value.

I'd say that technically it's a vector, but it's always in the same direction, so it's never going to change direction like curl in 3D.
 
  • #3
A curl is always the same type of beast in any number of dimensions. It's neither a vector nor a scalar; it's a bivector.

(Or a two-form; I'm not sure which. The point is that it's an intrinsically two-dimensional object.)

In 2D, the dual to a bivector is a scalar. In 3D, the dual to a bivector is a vector. Typically, students learn only about the vector, because bivectors are not typically taught.

So to answer your question: the curl in 2D is definitely not a vector. If you think of the 3D curl as a vector, you should think of the 2D curl as a scalar.
 
  • #4
So is wrong to say that [tex]\oint_{s}\vec{f}\cdot d\vec{s} = \iint_{A}\vec{\nabla}\times \vec{f} d^2A[/tex] ? Because in the left side we have an scalar and in the right side we have a vector and a scalar is not equal to a vector... BUT, if I use the full definition, the equation will be: [tex]\oint_{s}\vec{f}\cdot d\vec{s} = \iint_{A}\vec{\nabla}\times \vec{f} \cdot \hat{n} d^2A[/tex] However, we can't define a normal vector to xy plane because is assumed that we are in the operating in 2D... So, is necessary use the modulus in the equation: [tex]\oint_{s}\vec{f}\cdot d\vec{s} = \iint_{A}|\vec{\nabla}\times \vec{f}| d^2A[/tex] But which is the difference between ##|\vec{\nabla}\times \vec{f}|## and ##\vec{\nabla}\times \vec{f}##, in the pratice, none...
 
  • #5
Jhenrique said:
But which is the difference between ##|\vec{\nabla}\times \vec{f}|## and ##\vec{\nabla}\times \vec{f}##, in the pratice, none...

##|-1| \ne -1##.
 
  • #6
[itex]d^2A[/itex] should also be a bivector, and you should take the dot product between the two bivectors. This works equally well in 2D or 3D.

Notice how much more elegant this is: in particular, you never need to make a completely arbitrary choice between a "right hand rule" and a "left hand rule" (which would give you the same answer in the end).
 

1. Is curl in 2D a vector or a scalar?

The curl in 2D is a vector quantity.

2. How is the curl in 2D different from the curl in 3D?

The curl in 2D only has one component, while the curl in 3D has three components. This is because the curl in 2D is only concerned with rotation in a single plane, while the curl in 3D considers rotation in all three dimensions.

3. What is the physical significance of the curl in 2D?

The curl in 2D represents the amount of rotation or circulation of a vector field around a point in a two-dimensional plane. It is an important concept in fluid dynamics, as it helps to understand the flow of fluids.

4. How is the curl in 2D calculated?

The curl in 2D is calculated using the cross product of the gradient operator with the vector field. This results in a vector quantity with a magnitude and direction.

5. Can the curl in 2D be zero?

Yes, the curl in 2D can be zero if the vector field is conservative, meaning that it has a potential function. In this case, the vector field is not rotating around any point, and the curl is equal to zero everywhere.

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