Let's break down the problem step by step: --- ### **Step 1: Understanding the Problem** We are given: - A vector field: \(\mathbf{F} = x\mathbf{i} + y\mathbf{j} + 3z\mathbf{k}\) - The flux through a surface \(S\) is \(-\frac{1}{3} \frac{m^3}{s}\) - The surface \(S\) is the graph of \(f(x, y) = xy(2y + x - 2)\) over the triangle \(D\) with vertices \((0,0), (2,0), (0,1)\) - \(E\) is the solid between \(S\) and the \(xy\)-plane. We are to **explain why the flux of \(\mathbf{F}\) through the boundary of \(E\) is not a meaningful measure of the amount of fluid leaving \(E\).** --- ### **Step 2: Interpreting the Flux** Normally, the **flux** of a vector field \(\mathbf{F}\) through a closed surface (like the boundary of \(E\)) measures the net amount of "fluid" leaving the enclosed region per unit time. By the **Divergence Theorem**: \[ \iint_{\partial E} \mathbf{F} \cdot d\mathbf{A} = \iiint_{E} (\nabla \cdot \mathbf{F})\, dV \] #### **Calculate the Divergence:** \[ \nabla \cdot \mathbf{F} = \frac{\partial}{\partial x}(x) + \frac{\partial}{\partial y}(y) + \frac{\partial}{\partial z}(3z) = 1 + 1 + 3 = 5 \] So, the net flux out of any volume should be \(5\) times the volume, if this were a **physical fluid**. --- ### **Step 3: Checking Units** - \(\mathbf{F}\) has units of \(\frac{m^3}{s}\) - The flux has units of \(\frac{m^3}{s}\) - The divergence \((\nabla \cdot \mathbf{F})\) has units of \(\frac{1}{s}\) --- ### **Step 4: The Issue** #### **Why the Flux is Not a Meaningful Measure:** The divergence theorem connects the **net outflow** to the **rate of increase of fluid inside the region**. This is meaningful **only if** the vector field \(\mathbf{F}\) represents the velocity field of an **incompressible fluid** or the **rate of transport** of some quantity. However, in this case: - The divergence of \(\mathbf{F}\) is a **constant nonzero value** (\(5\)), which means the "fluid" is **being created at a constant rate everywhere** within \(E\). - Physically, this would only make sense if there were **sources throughout the volume**, creating fluid at a rate of \(5\) per unit volume per unit time. But in real fluids: - The net flux out of a region **should measure the amount of fluid leaving minus the amount entering**. - If the fluid is being created or destroyed inside the region (i.e., if \(\nabla \cdot \mathbf{F} \neq 0\)), the flux through the boundary doesn't represent just the fluid "leaving" due to flow—it also includes fluid "appearing" inside. #### **In summary:** The flux of \(\mathbf{F}\) through \(\partial E\) measures **not just fluid leaving** \(E\), but also fluid being **created inside** \(E\). --- ### **Step 5: Final Answer** #### **Explanation and Conclusion** > **The flux of \(\mathbf{F}\) through the boundary of \(E\) is not a meaningful measure of the amount of fluid leaving \(E\) because the divergence of \(\mathbf{F}\) is nonzero (\(\nabla \cdot \mathbf{F} = 5\)), meaning fluid is being created throughout the volume of \(E\) at a constant rate. Thus, the net flux through the boundary represents both the fluid leaving through the surface and the fluid being created inside \(E\), not just the fluid leaving. For the flux through the boundary to meaningfully represent only the fluid leaving, the vector field must be divergence-free (\(\nabla \cdot \mathbf{F} = 0\)), i.e., the fluid must be incompressible with no sources or sinks inside the region.** --- **In summary:** *The flux through the boundary of \(E\) is not just the fluid leaving \(E\), but also includes fluid created inside \(E\) due to the nonzero divergence.*