# Homework Help: If the divergence of a vector field is zero

1. Feb 10, 2007

1. The problem statement, all variables and given/known data
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. Relevant equations
Just the equations for divergence and curl. In TeX:
$$\nabla\cdot u=\frac{\partial u_x}{\partial x}+\frac{\partial u_y}{\partial y}+\frac{\partial u_z}{\partial z}$$
and the equivalent for curl.

3. The attempt at a solution
I really don't know at all how to find an answer.

Last edited by a moderator: Feb 11, 2007
2. Feb 11, 2007

### joob

The divergence of the curl of ANY vector is =0. You cant find that "vector" without some more information, eg boundary conditions.

3. Feb 11, 2007

### mjsd

4. Feb 11, 2007

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. Feb 11, 2007

### HallsofIvy

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:
$$\frac{\partial h}{\partial y}-\frac{\partial g}{\partial z}= u$$
$$\frac{\partial f}{\partial z}- \frac{\partial h}{\partial x}= v$$
$$\frac{\partial g}{\partial x}- \frac{\partial f}{\partial x}= w$$

Solve those for f, g, h,

6. Feb 11, 2007