1. **State the problem:** We want to compute the surface integral $$\iint_{S} \text{curl} \, \mathbf{F} \cdot d\mathbf{S}$$ where $$\mathbf{F}(x,y,z) = -y^{2} \mathbf{i} + x \mathbf{j} + z^{2} \mathbf{k}$$ and $$S$$ is the part of the paraboloid $$y = 4 - x^{2} - z^{2}$$ inside the cylinder $$x^{2} + z^{2} = 1$$. 2. **Use Stokes' Theorem:** Stokes' Theorem states: $$\iint_{S} \text{curl} \, \mathbf{F} \cdot d\mathbf{S} = \oint_{\partial S} \mathbf{F} \cdot d\mathbf{r}$$ where $$\partial S$$ is the boundary curve of surface $$S$$. 3. **Identify the boundary curve $$\partial S$$:** The boundary is where the paraboloid meets the cylinder: $$x^{2} + z^{2} = 1$$ and $$y = 4 - x^{2} - z^{2} = 4 - 1 = 3$$. So the boundary curve is the circle: $$x^{2} + z^{2} = 1, \quad y = 3$$. 4. **Parametrize the boundary curve:** Let $$t \in [0, 2\pi]$$, $$x = \cos t, \quad z = \sin t, \quad y = 3$$. 5. **Compute $$d\mathbf{r}$$:** $$d\mathbf{r} = \frac{d\mathbf{r}}{dt} dt = (-\sin t \mathbf{i} + 0 \mathbf{j} + \cos t \mathbf{k}) dt$$. 6. **Evaluate $$\mathbf{F}$$ on the boundary:** $$\mathbf{F}(x,y,z) = -y^{2} \mathbf{i} + x \mathbf{j} + z^{2} \mathbf{k}$$ At boundary: $$y=3 \Rightarrow y^{2} = 9$$, $$x = \cos t$$, $$z = \sin t$$, so $$\mathbf{F} = -9 \mathbf{i} + \cos t \mathbf{j} + \sin^{2} t \mathbf{k}$$. 7. **Compute the line integral:** $$\mathbf{F} \cdot d\mathbf{r} = (-9)(-\sin t) + (\cos t)(0) + (\sin^{2} t)(\cos t) dt = 9 \sin t + \sin^{2} t \cos t \, dt$$. 8. **Integrate over $$t$$ from 0 to $$2\pi$$:** $$\int_{0}^{2\pi} \left(9 \sin t + \sin^{2} t \cos t \right) dt = \int_{0}^{2\pi} 9 \sin t \, dt + \int_{0}^{2\pi} \sin^{2} t \cos t \, dt$$. 9. **Evaluate each integral:** - $$\int_{0}^{2\pi} 9 \sin t \, dt = 9[-\cos t]_{0}^{2\pi} = 9(-\cos 2\pi + \cos 0) = 9(-1 + 1) = 0$$. - For $$\int_{0}^{2\pi} \sin^{2} t \cos t \, dt$$, use substitution: Let $$u = \sin t$$, then $$du = \cos t dt$$. When $$t=0$$, $$u=0$$; when $$t=2\pi$$, $$u=0$$. So integral becomes: $$\int_{u=0}^{0} u^{2} du = 0$$. 10. **Sum of integrals:** $$0 + 0 = 0$$. **Final answer:** $$\boxed{0}$$ The surface integral of the curl of $$\mathbf{F}$$ over the paraboloid portion inside the cylinder is zero.