Integration against divergence free vector fields Let $\chi:\Omega\to \mathbb{R}^n$ be a vector field on a bounded, smooth domain $\Omega \subset \mathbb{R}^n$. Assume that for any divergence free vector field $\eta:\Omega \to \mathbb{R}^n$ we have 
$
\int_\Omega \eta \cdot \chi =0
$
Does this imply that $\chi = 0$ ?
 A: This does not imply that $\chi = 0$. What can be said is the following:
Let $\chi \colon \overline{\Omega} \rightarrow \mathbb{R}^n$ be a vector field on a bounded, smooth domain $\overline{\Omega} \subset \mathbb{R}^n$. Then, 
$$ \int_{\Omega} \eta \cdot \chi \, d\Omega = 0 $$
for all smooth divergence free vector fields $\eta \colon \overline{\Omega} \rightarrow \mathbb{R}^n$ if and only if there exist a smooth function $\phi \colon \overline{\Omega} \rightarrow \mathbb{R}$ such that $\chi = \nabla\phi$ and $\left. \phi \right|_{\partial \Omega} = 0$.
Proof: Assume that $\chi = \nabla \phi$ with $\left. \phi \right|_{\partial \Omega} = 0$. Then, using Green's identity for integration by parts we have
$$ \int_{\Omega} (\nabla \phi) \cdot \eta \, d\Omega = \int_{\partial \Omega} \phi (\eta \cdot \hat{N}) \, dS - \int_{\Omega} \phi (\nabla \cdot \eta) \, d\Omega = 0 $$
for all divergence free vector fields $\eta$. For the other direction, given $\chi$ we can solve the Poisson problem $\Delta \phi = \nabla \cdot \chi$ on $\Omega$ with boundary condition $\left. \phi \right|_{\partial \Omega} = 0$. Such a solution satisfies $\Delta \phi = \nabla \cdot (\nabla \phi) = \nabla \cdot \chi$ which implies that $\nabla \cdot (\nabla \phi - \chi) = 0$. Thus, we can write $\nabla \phi = \chi + F$ with $F = \nabla \phi - \chi$ and $\nabla \cdot F = 0$. Taking $\eta = F$, we see that
$$ 0 = \int_{\Omega} \eta \cdot \chi \, d\Omega = \int_{\Omega} F \cdot  (\nabla \phi - F) \, d\Omega = \int_{\Omega} \nabla \phi \cdot F \, d\Omega - \int_{\Omega} F \cdot F \, d \Omega = -\int_{\Omega} F \cdot F \, d\Omega.$$
Thus, $F \equiv 0$ and $\nabla \phi = \chi$ with $\left. \phi \right|_{\partial \Omega} = 0$.

Remark: This shows that the orthogonal complement to the subspace of smooth divergent free vector fields on $\overline{\Omega}$ is given by the space of "grounded gradients" (gradients of smooth functions that vanish on the boundary of $\Omega$) where the inner product on the space of vector fields is given by the formula
$$ \left< X, Y \right> = \int_{\Omega} X \cdot Y \, d\Omega. $$
