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I read about Helmholtz theorem on the Griffiths (Introduction to electrodynamics) and in the appendix B it shows how to use the Helmholtz theorem to determine the field F starting from the potential. The book says that:

Suppose we are told that the divergence of a vector function F(r) is a specified scalar function D(r): $\nabla \cdot \textbf{F}=D(r)$

and the curl of F(r) is a specified vector function C(r): $\nabla \times \textbf{F}=\textbf{C}$

For consistency, C must be divergenceless, $\nabla \cdot \textbf{C}=0$ because the divergence of a curl is always zero. Question: can we, on the basis of this information, determine the function F? If D(r) and C(r) go to zero sufficiently rapidly at infinity, the answer is yes, as I will show by explicit construction. I claim that

$\textbf{F}=-\nabla U + \nabla \times \textbf{W}$


$U(\textbf{r})=\frac{1}{4\pi}\int \frac{D(\textbf{r'})}{\textbf{r-r'}}d\tau'$


$\textbf{W(r)}=\frac{1}{4\pi}\int \frac{\textbf{C(r'})}{\textbf{r-r'}}d\tau'$

Can someone show me how to obtain the potentials formulas written above?

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They are the Green's Functions for the Helmholtz equation – Jorge Jun 17 '13 at 14:56

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