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Helmholtz theorem states that given a smooth vector field $\mathbf{H}$, there are a scalar field $\phi$ and a vector field $\mathbf{G}$ such that

$\mathbf{H}=\nabla \phi +\nabla \times \mathbf{G}$

and

$\nabla \mathbf{\cdot G}=0$

Is this decomposition unique? That is, given $\mathbf{H}$, are the fields $\phi$, $\mathbf{G}$ satisfying the above equations unique?

Edit: Unique, up to an additive constant.

Thanks

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    cross-post: http://physics.stackexchange.com/q/10522/30642013-05-30

1 Answers 1

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The decomposition is not unique without further conditions. You can add linear terms to $\phi$ and $\mathbf G$ that yield constant contributions to $\mathbf H$ that cancel:

$ \begin{eqnarray} \phi &\to& \phi + z\;, \\ \nabla\phi &\to& \nabla\phi + \mathbf e_z\;, \\ \mathbf G &\to& \mathbf G + \frac{1}{2}(y\mathbf e_x-x\mathbf e_y)\;, \\ \nabla\times\mathbf G &\to& \nabla\times\mathbf G - \mathbf e_z\;. \end{eqnarray} $

However, I think that if you impose suitable conditions that the fields decay at infinity, the decomposition should be unique.

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    Thanks. Yes, I realize now that boundary conditions are essential. After boundary conditions have been specified, the resolution is unique.2011-05-29