Is the curl of a div. free vector field perpendicular to the field?

In summary, the conversation is about the divergence-free vector field and its curl being perpendicular to the field. The speaker is having trouble proving this to themselves and presents an example to illustrate their point.
  • #1
Wuberdall
34
0
Hi PF-members.

My intuition tells me that: Given a divergence free vector field [itex] \mathbf{F} [/itex], then the curl of the field will be perpendicular to field.
But I'm having a hard time proving this to my self.

I'know that : [itex] \nabla\cdot\mathbf{F} = 0 \hspace{3mm} \Rightarrow \hspace{3mm} \exists\mathbf{A}: \mathbf{F} = \nabla\times\mathbf{A} [/itex]

Therefore : [itex] \mathbf{F}\cdot(\nabla\times\mathbf{F}) = 0 \hspace{3mm} \Rightarrow \hspace{3mm} [\nabla\times\mathbf{A}]\cdot[\nabla\times(\nabla\times\mathbf{A})] = 0 [/itex]

But I can't prove that this actually equals zero... Please help!
 
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  • #2
Off the top of my head (as in, there are probably less artificial examples):

Let [itex]\mathbf{F} = (x-z,y-z,-2z)[/itex]. As [itex]\nabla \cdot \mathbf{F} = 1+1-2=0[/itex], the field is divergence-free. However, [itex]\nabla \times \mathbf{F} = (1,-1,0)[/itex], so, clearly, [itex]\mathbf{F}\cdot (\nabla \times \mathbf{F})\not{\equiv}0[/itex] and by contradiction, your intuition seems to be wrong.
 
Last edited:
  • #3
Gracias.
 

1. What is a divergence-free vector field?

A divergence-free vector field is a vector field in which the divergence, or the measure of the outflow of a vector field from a given point, is equal to zero. In other words, the vector field has no sources or sinks at any point in space.

2. What is the curl of a vector field?

The curl of a vector field is a vector operation that describes the rotation or circulation of a vector field at a given point. It is represented by the vector cross product of the del operator and the vector field itself. The magnitude and direction of the curl at a point indicate the amount of rotation or circulation of the vector field at that point.

3. How do you determine if a vector field is divergence-free?

A vector field is divergence-free if the divergence at every point in space is equal to zero. This can be determined by taking the partial derivatives of the vector field with respect to each of the three spatial dimensions and checking if the resulting sums are equal to zero at every point.

4. What does it mean for the curl to be perpendicular to the vector field?

If the curl of a vector field is perpendicular to the vector field itself, it means that the vector field is solenoidal, or that it has no net rotation or circulation at any point. This is only true for divergence-free vector fields.

5. Why is it important for the curl to be perpendicular to a div. free vector field?

This is important because it allows for the representation of a vector field as the gradient of a scalar function, known as a potential function. This simplifies calculations and makes it easier to analyze the behavior of the vector field. Additionally, the perpendicular relationship between the curl and the vector field is a fundamental property of divergence-free vector fields and is essential in many applications in physics and engineering.

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