In Ex 6.7 part (a) of Stone-Goldbart, we are required to use the Fredholm condition for the existence of solution for a Neumann-Poisson problem to derive the Helmholtz decomposition. Given \begin{align*} -\nabla^2 \varphi &= f \\ \hat{n}\cdot\nabla\varphi &= g \end{align*} and if \begin{align*} \int_{\Omega} f d^nr + \int_{\partial \Omega} g ds = 0 \end{align*} holds over $\Omega$, then $\varphi$ exists and it is unique. Show that any smooth vector field $\vec{u}$ can be decomposed into $\vec{u} = \vec{v} + \nabla \phi$ where $\phi$ is unique up to a constant, and $\vec{v} \cdot n = 0$ along $\partial \Omega$, $\nabla \cdot \vec{v} = 0$ throughout $\Omega$. I am having some difficulties to sort out the logics. I tried to use Divergence Theorem to write down \begin{align*} \int_{\Omega} -\nabla \cdot \vec{u} d^nr + \int_{\partial \Omega} \hat{n} \cdot \vec{u} ds = 0 \end{align*} and subtract the Fredholm condition from it to obtain \begin{align*} \int_{\Omega} -\nabla \cdot (\vec{u} - \nabla\varphi) d^nr + \int_{\partial \Omega} \hat{n}\cdot(\vec{u} - \nabla \varphi) ds = 0 \end{align*} I attempt to rewrite $\vec{u} - \nabla\varphi = \vec{v}$, hence \begin{align*} \int_{\Omega} -\nabla \cdot \vec{v} d^nr + \int_{\partial \Omega} \hat{n} \cdot \vec{v} ds = 0 \end{align*} Now I am unsure if I can claim that I can select at will $\vec{v} \cdot n = 0$ and $\nabla \cdot \vec{v} = 0$, and how the uniqueness of $\vec{v}$ (and hence $\varphi$ or $\phi$) can hold. Any help would be appreciated! (Much appreciated if parts (b) and (c) can be addressed as well!)
Note: I am aware of this question, but it is centered at the uniqueness of the Neumann problem itself. And I want to how this can translate to the decomposition by the approach suggested by the textbook.