Word Count : 551

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To solve this using the Divergence Theorem, we will follow these steps: ### Step 1: Recall the Divergence Theorem The Divergence Theorem states: \[ \iint_{S} \vec{F} \cdot d\vec{S} = \iiint_{V} (\nabla \cdot \vec{F}) \, dV \] where: - \( \vec{F} = y^{2}z \hat{i} + y^{3} \hat{j} + xz \hat{k} \) - \( S \) is the closed surface of the cube defined by \( -1 \leq x \leq 1 \), \( -1 \leq y \leq 1 \), and \( 0 \leq z \leq 2 \) - \( V \) is the volume enclosed by the surface \( S \) ### Step 2: Compute the Divergence of \( \vec{F} \) The divergence of a vector field \( \vec{F} = P\hat{i} + Q\hat{j} + R\hat{k} \) is given by: \[ \nabla \cdot \vec{F} = \frac{\partial P}{\partial x} + \frac{\partial _______ _________ ________ ______ ___ ____ ______ _________ ____.
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