A fluid has density 4 and velocity field v = -yi + xj + 3zk. Find the rate of flow outward through the sphere x^2 + y^2 + z^2 = 16
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Step 1: Calculate the divergence of the velocity field, which is given as \( \nabla \cdot \textbf{v} = \frac{\partial (-y)}{\partial x} + \frac{\partial x}{\partial y} + \frac{\partial 3z}{\partial z} = -1 + 1 + 3 = 3 \). Show more…
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