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If the divergence of a vector field is zero

  • Thread starter adamabel
  • Start date
7
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1. 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?


2. 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.


3. The Attempt at a Solution
I really don't know at all how to find an answer.
 
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Answers and Replies

13
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The divergence of the curl of ANY vector is =0. You cant find that "vector" without some more information, eg boundary conditions.
 
mjsd
Homework Helper
725
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1. 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.
 
7
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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?
 
HallsofIvy
Science Advisor
Homework Helper
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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,
 
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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.
 

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