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}.