Divergence of the curl problem question

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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 [tex]{\nabla}{\cdot}{\vec{V}}=0[/tex] and try to find any vector, U, for which [tex]{\vec{V}}={\nabla}{\cdot}{\vec{U}}[/tex]
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, [itex]{\vec{U}}[/itex], for which, say, [itex]{\vec{U_z}}=0[/itex]

The Attempt at a Solution



I know that for [tex]{\nabla}{\cdot}{\vec{V}}=0[/tex] we have to have for example, a vector field such as: [tex]{\vec{V}}={yz}\hat{x}+{xz}\hat{y}+{xy}\hat{z}[/tex] so that when we do [tex]{\nabla}{\cdot}{\vec{V}}=0[/tex]. 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).
 
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Answers and Replies

  • #2
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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 [tex]{\nabla}{\cdot}{\vec{V}}=0[/tex] and try to find any vector, U, for which [tex]{\vec{V}}={\nabla}{\cdot}{\vec{U}}[/tex]
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, [itex]{\vec{U}}[/itex], for which, say, [itex]{\vec{U_z}}=0[/itex]

The Attempt at a Solution



I know that for [tex]{\nabla}{\cdot}{\vec{V}}=0[/tex] we have to have for example, a vector field such as: [tex]{\vec{V}}={yz}\hat{x}+{xz}\hat{y}+{xy}\hat{z}[/tex] so that when we do [tex]{\nabla}{\cdot}{\vec{V}}=0[/tex]. 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
[tex]\nabla \cdot \vec{V} = 0[/tex]
for any particular vector field [itex]\vec{V}[/itex], then you can always find a vector field [itex]\vec{U}[/itex] such that
[tex]\nabla \times \vec{U} = \vec{V}[/tex]
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 [itex]\vec{A}\cdot\vec{B}=0[/itex], then there is a vector [itex]\vec{C}[/itex] such that [itex]\vec{A}\times\vec{C}=\vec{B}[/itex].
 
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