- #1

- 30

- 1

## 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).

Last edited by a moderator: