Answer
See proof
Work Step by Step
Let $\vec{F}(x, y, z) = a\hat{i} + b\hat{j} + c\hat{k}$ be a constant vector field, and let $\sigma$ be the surface of the solid $G$. Using the Divergence Theorem, we have: \[ \Phi = \iint_{\sigma} \vec{F} \cdot \vec{n} \, ds = \iint_{G} \nabla \cdot \vec{F} \, dV, \] where \[ \nabla \cdot \vec{F} = \frac{\partial}{\partial x}a + \frac{\partial}{\partial y}b + \frac{\partial}{\partial z}c = 0, \] because $\vec{F}$ is a constant vector field (i.e., $a$, $b$, and $c$ are constants). Then, \[ \Phi = \iiint_{G} (0) \, dV = 0. \] Therefore, the flux $\Phi$ is equal to zero.
