To solve the problem, we utilize Gauss's Divergence Theorem, which relates the flux of a vector field $\mathbf{F}$ through the surface $S$ to the divergence of $\mathbf{F}$ over the volume $W$ enclosed by $S$. Specifically:

$\iint_S \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_W \text{div}(\mathbf{F}) \, dV.$

Step-by-step solution:

1. Understand the geometry:

   • $S$ is the surface of a half-cylinder $x^2 + y^2 = 1$ (for $x \geq 0$), with $0 \leq z \leq 1$.
   • $W$ is the solid region enclosed by this surface, including the curved side, the flat circular base ($z = 0$), and the top disk ($z = 1$).

2. Vector field $\mathbf{F}$: The vector field is given by: $\mathbf{F}(x, y, z) = zy \, \mathbf{i}.$ Since $\mathbf{F}$ has no $\mathbf{j}$ or $\mathbf{k}$ components, $\mathbf{F}$ is horizontal.

3. Divergence of $\mathbf{F}$: The divergence is computed as: $\text{div}(\mathbf{F}) = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z},$ where $F_1 = zy$, $F_2 = 0$, $F_3 = 0$. Thus: $\text{div}(\mathbf{F}) = \frac{\partial (zy)}{\partial x} + \frac{\partial (0)}{\partial y} + \frac{\partial (0)}{\partial z} = 0.$

4. Given condition: Despite the above divergence being zero for this $\mathbf{F}$, the problem explicitly states: $\iiint_W \text{div}(\mathbf{F}) \, dV = 4.$ This indicates a net flux of $4$ across all boundary surfaces of $S$.

5. Flux through the curved surface: The total flux is distributed across:

   • The curved side of $S$,
   • The circular disk at $z = 0$,
   • The circular disk at $z = 1$.

   By symmetry and the horizontal nature of $\mathbf{F}$, there is no contribution to the flux from the top and bottom disks. Hence, the flux through $S$ (the curved side) equals the total flux: $\boxed{4}.$

Let me know if you'd like a deeper explanation! Here are related questions to expand your understanding:

Related Questions:

1. How does Gauss's Divergence Theorem simplify evaluating flux integrals?
2. Why does the flux through the flat surfaces vanish in this problem?
3. How would the solution change if $\mathbf{F}$ had a nonzero $z$-component?
4. What role does symmetry play in simplifying the evaluation of flux?
5. Can you derive the divergence of a more complex vector field, e.g., $\mathbf{F}(x, y, z) = xz \mathbf{i} + yz \mathbf{j} + xy \mathbf{k}$?

Tip:

Always check if the vector field has symmetry or special properties (like being divergence-free) to simplify the flux or divergence calculations!