= Solution
Writing $r^2=x^2+y^2+z^2$, the <gradient> is $\mathbf v=2e^{r^2}(x,y,z)$. Its <divergence>, equivalently the <Laplacian> of $\phi$, is
$$
\nabla\cdot\mathbf v=(6+4r^2)e^{r^2}.
$$
On the unit <sphere> this <scalar> is the constant $10e$. The lower <hemisphere> has <surface area> $2\pi$, so
$$
\boxed{\int_S(\nabla\cdot\mathbf v)\,dS=20\pi e}.
$$
This is a <scalar> <surface integral> of the divergence, not the electric- or vector-field flux <integral>. There is no normal component to insert.
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