1. **Problem Statement:** Calculate the circulation of the vector field $\mathbf{F} = (yz + x, xz - y, xy + z)$ around the triangular loop with vertices $A(1,0,0)$, $B(0,1,0)$, and $C(0,0,1)$ using Stokes' Theorem. 2. **Stokes' Theorem:** Stokes' Theorem relates the circulation of a vector field around a closed curve $C$ to the surface integral of the curl of the field over a surface $S$ bounded by $C$: $$\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n} \, dS$$ where $\mathbf{n}$ is the unit normal vector to the surface. 3. **Calculate the curl $\nabla \times \mathbf{F}$:** $$\nabla \times \mathbf{F} = \left( \frac{\partial}{\partial y}(xy + z) - \frac{\partial}{\partial z}(xz - y), \frac{\partial}{\partial z}(yz + x) - \frac{\partial}{\partial x}(xy + z), \frac{\partial}{\partial x}(xz - y) - \frac{\partial}{\partial y}(yz + x) \right)$$ Calculate each component: - $\frac{\partial}{\partial y}(xy + z) = x$ - $\frac{\partial}{\partial z}(xz - y) = x$ - $\frac{\partial}{\partial z}(yz + x) = y$ - $\frac{\partial}{\partial x}(xy + z) = y$ - $\frac{\partial}{\partial x}(xz - y) = z$ - $\frac{\partial}{\partial y}(yz + x) = z$ So, $$\nabla \times \mathbf{F} = (x - x, y - y, z - z) = (0, 0, 0)$$ 4. **Interpretation:** The curl of $\mathbf{F}$ is zero everywhere, so the surface integral of the curl over any surface bounded by $C$ is zero. 5. **Conclusion:** By Stokes' Theorem, $$\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n} \, dS = 0$$ **Final answer:** The circulation of the data packets around the triangular loop is $0$.