1. **Problem Statement:** Compute the circulation of the heat/data flow vector field $ \vec{F} = (yz^2, xz^2, x^2 + y^2) $ along the boundary of the circular region defined by $ x^2 + y^2 \leq 4 $ using Stokes' Theorem. 2. **Recall Stokes' Theorem:** Stokes' Theorem relates the circulation of a vector field $ \vec{F} $ around a closed curve $ C $ to the surface integral of the curl of $ \vec{F} $ over a surface $ S $ bounded by $ C $: $$ \oint_C \vec{F} \cdot d\vec{r} = \iint_S (\nabla \times \vec{F}) \cdot \vec{n} \, dS $$ where $ \vec{n} $ is the unit normal vector to the surface. 3. **Parameterize the surface:** The surface $ S $ is the disk $ x^2 + y^2 \leq 4 $ in the plane $ z=0 $. 4. **Evaluate the curl $ \nabla \times \vec{F} $:** Given $ \vec{F} = (F_x, F_y, F_z) = (yz^2, xz^2, x^2 + y^2) $, compute: $$ \nabla \times \vec{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}, \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}, \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) $$ Calculate each component: - $ \frac{\partial F_z}{\partial y} = \frac{\partial}{\partial y}(x^2 + y^2) = 2y $ - $ \frac{\partial F_y}{\partial z} = \frac{\partial}{\partial z}(xz^2) = 2xz $ - $ \frac{\partial F_x}{\partial z} = \frac{\partial}{\partial z}(yz^2) = 2yz $ - $ \frac{\partial F_z}{\partial x} = \frac{\partial}{\partial x}(x^2 + y^2) = 2x $ - $ \frac{\partial F_y}{\partial x} = \frac{\partial}{\partial x}(xz^2) = z^2 $ - $ \frac{\partial F_x}{\partial y} = \frac{\partial}{\partial y}(yz^2) = z^2 $ Thus, $$ \nabla \times \vec{F} = (2y - 2xz, 2yz - 2x, z^2 - z^2) = (2y - 2xz, 2yz - 2x, 0) $$ 5. **Evaluate curl on the surface $ z=0 $:** Substitute $ z=0 $: $$ \nabla \times \vec{F} = (2y - 0, 0 - 2x, 0) = (2y, -2x, 0) $$ 6. **Normal vector to the surface:** Since the surface is in the plane $ z=0 $, the unit normal vector pointing upward is: $$ \vec{n} = (0, 0, 1) $$ 7. **Dot product $ (\nabla \times \vec{F}) \cdot \vec{n} $:** $$ (2y, -2x, 0) \cdot (0, 0, 1) = 0 $$ 8. **Surface integral:** $$ \iint_S (\nabla \times \vec{F}) \cdot \vec{n} \, dS = \iint_S 0 \, dS = 0 $$ 9. **Conclusion:** The circulation of $ \vec{F} $ along the boundary of the circular region is: $$ \boxed{0} $$ This means there is no net circulation of heat/data flow along the boundary of the disk.