If the divergence of a vector field is zero

In summary, you need a vector field that has divergence equal to 0 in order to find its curl, but you don't know how to find that vector field.
  • #1
adamabel
7
0

Homework Statement


If the divergence of a vector field is zero, I know that that means that it is the curl of some vector. How do I find that vector?


Homework Equations


Just the equations for divergence and curl. In TeX:
[tex]\nabla\cdot u=\frac{\partial u_x}{\partial x}+\frac{\partial u_y}{\partial y}+\frac{\partial u_z}{\partial z}[/tex]
and the equivalent for curl.


The Attempt at a Solution


I really don't know at all how to find an answer.
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
The divergence of the curl of ANY vector is =0. You can't find that "vector" without some more information, eg boundary conditions.
 
  • #3
adamabel said:

Homework Statement


If the divergence of a vector field is zero, I know that that means that it is the curl of some vector. How do I find that vector?
the statement: [tex]\nabla\cdot(\nabla\times A)=0[/tex] is true for all vector field A. So without any additional info, you just have an arbitrary vector field.
 
  • #4
So when a problem gives a vector field where it's divergence is zero, and it asks to find a vector field such that the curl of the vector field is the given vector field, I can just choose any vector field?
 
  • #5
No, those responses were to what you had posted before- that all you knew about the vector field was that its divergence was equal to 0. You did not say you were given a vector field that happened to have divergence equal to 0!

If you are given a vector field, say, u(x,y,z)i+ v(x,y,z)j+ w(x,y,z)k with divergence 0, Then write out the formula for curl of a vector field and set the components equal:
[tex]\frac{\partial h}{\partial y}-\frac{\partial g}{\partial z}= u[/tex]
[tex]\frac{\partial f}{\partial z}- \frac{\partial h}{\partial x}= v[/tex]
[tex]\frac{\partial g}{\partial x}- \frac{\partial f}{\partial x}= w[/tex]

Solve those for f, g, h,
 
  • #6
I already knew that; I suppose I just didn't write it out clearly enough. But what was confusing me was how to solve for those. It seems like that is a system of PDEs, and I have no idea how to solve those.
 

1. What does it mean for the divergence of a vector field to be zero?

When the divergence of a vector field is zero, it means that the net flow of the vector field into any closed surface is equal to the net flow out of that surface. In other words, the vector field has no sources or sinks within the volume enclosed by the surface.

2. What are some real-life examples of vector fields with zero divergence?

Some common examples of vector fields with zero divergence include electrostatic fields, gravitational fields, and fluid flows that are both incompressible and irrotational.

3. How is the divergence of a vector field related to its curl?

The divergence and the curl of a vector field are related by the divergence theorem, which states that the outward flux of the curl of a vector field through a closed surface is equal to the circulation of the vector field around the boundary of that surface. In other words, the divergence of a vector field represents its tendency to spread out, while the curl represents its rotational behavior.

4. Can a vector field with zero divergence still have non-zero curl?

Yes, it is possible for a vector field to have both zero divergence and non-zero curl. This occurs when the vector field has regions of both expansion and rotation, such as in a vortex flow.

5. How is the concept of divergence used in practical applications?

The concept of divergence is used in many practical applications, such as fluid dynamics, electromagnetism, and computer graphics. For example, in fluid dynamics, the divergence of a velocity field can be used to determine the strength and location of sources or sinks within a fluid flow. In computer graphics, the divergence of a vector field can be used to create realistic-looking smoke or fire effects.

Similar threads

  • Calculus and Beyond Homework Help
Replies
8
Views
828
Replies
33
Views
3K
  • Calculus and Beyond Homework Help
Replies
4
Views
765
  • Calculus and Beyond Homework Help
Replies
4
Views
913
  • Calculus and Beyond Homework Help
Replies
1
Views
2K
Replies
4
Views
604
  • Calculus and Beyond Homework Help
Replies
2
Views
960
  • Calculus and Beyond Homework Help
Replies
3
Views
837
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
635
Back
Top