Answer
$\phi = 0$
Work Step by Step
Let $\vec{F} = 5\hat{j} + 7\hat{k}$ and the spherical surface $x^2 + y^2 + z^2 = 1$ with unit normals $\hat{n} = \frac{\vec{r}}{|\vec{r}|} = \vec{r}^{\,}$. The flux of $\vec{F}$ across the surface $\sigma$ is given by \[ \begin{aligned} \phi &= \iint_{\sigma} \vec{F} \cdot \hat{n} \, ds \\ &= \iint_{\sigma} (5\hat{j} + 7\hat{k}) \cdot \hat{n} \, ds \\ &= \iint_{\sigma} (5\sin\theta\sin\phi + 7\cos\phi) \, ds \\ &= \int_0^{2\pi} \int_0^{\pi} (5\sin\theta\sin\phi + 7\cos\phi) \, d\theta d\phi \\ &= 0. \end{aligned} \] Now, using the Divergence Theorem, where \[ \begin{aligned} \nabla \cdot \vec{F} &= \nabla \cdot (5\hat{j} + 7\hat{k}) \\ &= 0, \end{aligned} \] the flux result is \[ \begin{aligned} \phi &= \iiint_{G} \nabla \cdot \vec{F} \, dV \\ &= 0. \end{aligned} \] Finally, the flux is \[ \text{flux} = \phi = 0. \]
