Answer
$\phi = 12$
Work Step by Step
Given Vector Field: \[ \mathbf{F}(x, y, z) = (x^2 + y)\mathbf{i} + z^2\mathbf{j} + (e^y - z)\mathbf{k} \] Calculation: 1. Calculate the divergence of \(\mathbf{F}\): \[ \nabla \cdot \mathbf{F} = 2x - 1 \] 2. Set up the triple integral for the flux across the specified rectangular solid: \[ \iiint_G (\nabla \cdot \mathbf{F}) \, dV \] where \(G\) is the region defined by \(0 \leq x \leq 3\), \(0 \leq y \leq 1\), and \(0 \leq z \leq 2\). 3. Evaluate the triple integral: \[ \begin{aligned} \phi &= \iiint_G (2x - 1) \, dV \\ &= \left[ \frac{2x^2}{2} - x \right]_{0}^{3} \cdot (1 - 0) \cdot (2 - 0) \\ &= 6 \cdot 2 \\ &= 12 \end{aligned} \] Result: The flux of the vector field \(\mathbf{F}\) across the specified rectangular solid and coordinate planes is \(\phi = 12\).
