Suppose $V\subset \mathbb{R}^3$ be non-empty and at least twice differentiable (Smooth) and let $S$ be the surface that encloses $V$ (for example a sphere). Let $\textbf{F}\in \mathbb{R}^3$ be a smooth vector field for all space. Let $\textbf{n}$ denote the normal to the surface $S$. It is well known using Helmhotz decompostion that we can decompose $\textbf{F}$ into two vector fields in $V$; $$\textbf{F} = \nabla \sigma + \nabla \times \Gamma$$ where $\nabla \sigma$ is called the irrotatioal part and $\nabla \times \Gamma$ the selonodial part of $\textbf{F}$. This decomposositon is known not to be unique. I want to know if the class of all solutions are related. For example take $\sigma\_1$, $\sigma\_2$, $\Gamma\_1$, and $\Gamma\_2$ so that $$\textbf{F} = \nabla \sigma\_1 + \nabla \times \Gamma\_1 = \nabla \sigma\_2 + \nabla \times \Gamma\_2$$ and then $$ \nabla (\sigma\_1- \sigma\_2) + \nabla \times (\Gamma\_1- \Gamma\_2) = \textbf{0}.$$ This should happen only if both these vector fields vanish. Or $$ \nabla \times (\Gamma\_1- \Gamma\_2) = \textbf{0}$$ $$ \nabla (\sigma\_1- \sigma\_2) = \textbf{0}.$$ Is there anything we can say about the solutions (i,e are they linear independent or multiples of one another).



Are all Helmholtz decompositions related?
**Q:** *How are two Helmholtz decompositions related?*


**A:** The scalar fields differ by a *harmonic function*.


Starting from a first decomposition $\sigma\_1,\Gamma\_1$, you can construct a second one by adding to $\sigma\_1$ a harmonic function $h$,
$$\sigma\_2=\sigma\_1+h,\;\;\text{with}\;\;\nabla^2 h=0.$$
Then the matching $\Gamma\_2$ is determined by the equation
$$\nabla\times(\Gamma\_1-\Gamma\_2)=\nabla h.$$


