# If the divergence of a vector field is zero

## 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:
$$\nabla\cdot u=\frac{\partial u_x}{\partial x}+\frac{\partial u_y}{\partial y}+\frac{\partial u_z}{\partial z}$$
and the equivalent for curl.

## The Attempt at a Solution

I really don't know at all how to find an answer.

Last edited by a moderator:

joob
The divergence of the curl of ANY vector is =0. You can't find that "vector" without some more information, eg boundary conditions.

Homework Helper

## 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: $$\nabla\cdot(\nabla\times A)=0$$ is true for all vector field A. So without any additional info, you just have an arbitrary vector field.

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?

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