Divergence of the curl problem question

[galactic]

Homework Statement

if a vector can be written as the curl of another vector, its divergence vanishes. Can you justify the statement: "any vector field whose divergence vanishes identically can be written as the curl of some other vector"?

Homework Equations

Prove this by construction. Let $${\nabla}{\cdot}{\vec{V}}=0$$ and try to find any vector, U, for which $${\vec{V}}={\nabla}{\cdot}{\vec{U}}$$
This amounts to showing that you have enough freedom to pick as components of \vec{U}, functions which satisfy some simple differetial equations. Simplify somewhat by trying to find a vector, ${\vec{U}}$, for which, say, ${\vec{U_z}}=0$

The Attempt at a Solution

I know that for $${\nabla}{\cdot}{\vec{V}}=0$$ we have to have for example, a vector field such as: $${\vec{V}}={yz}\hat{x}+{xz}\hat{y}+{xy}\hat{z}$$ so that when we do $${\nabla}{\cdot}{\vec{V}}=0$$. I'm confused if the problem is just asking to prove the divergence of the curl is equal to 0 which I have already done a few homework problems ago or if its asking for something different here...because I'm confused by the hint that it's giving me (in relevant equations).

 

[slider142]

Homework Statement

if a vector can be written as the curl of another vector, its divergence vanishes. Can you justify the statement: "any vector field whose divergence vanishes identically can be written as the curl of some other vector"?

Homework Equations

Prove this by construction. Let $${\nabla}{\cdot}{\vec{V}}=0$$ and try to find any vector, U, for which $${\vec{V}}={\nabla}{\cdot}{\vec{U}}$$
This amounts to showing that you have enough freedom to pick as components of \vec{U}, functions which satisfy some simple differetial equations. Simplify somewhat by trying to find a vector, ${\vec{U}}$, for which, say, ${\vec{U_z}}=0$

The Attempt at a Solution

I know that for $${\nabla}{\cdot}{\vec{V}}=0$$ we have to have for example, a vector field such as: $${\vec{V}}={yz}\hat{x}+{xz}\hat{y}+{xy}\hat{z}$$ so that when we do $${\nabla}{\cdot}{\vec{V}}=0$$. I'm confused if the problem is just asking to prove the divergence of the curl is equal to 0 which I have already done a few homework problems ago or if its asking for something different here...because I'm confused by the hint that it's giving me (in relevant equations).

I believe there is a typo in your relevant equations. They want you to show that if
$$\nabla \cdot \vec{V} = 0$$
for any particular vector field $\vec{V}$, then you can always find a vector field $\vec{U}$ such that
$$\nabla \times \vec{U} = \vec{V}$$
They want you to construct a vector field U by x, y and z components that satisfies the second equation, using the equation for the divergence of V being 0 as the only given fact.
This is actually similar to a proof you may have done earlier: in 3-dimensional space, if $\vec{A}\cdot\vec{B}=0$, then there is a vector $\vec{C}$ such that $\vec{A}\times\vec{C}=\vec{B}$.