Evaluate the surface integral ∬_S F · n dS, where F = x i + y j + z k and S is the closed sphere x^2 + y^2 + z^2 = R^2 with outward unit normal n.
Answer and explanation
Correct answer: D. 4π R^3
- Step 1: Apply Gauss Divergence Theorem: ∬_S F · n dS = ∭_V (div F) dV.
- Step 2: div F = ∂(x)/∂x + ∂(y)/∂y + ∂(z)/∂z = 1 + 1 + 1 = 3.
- Step 3: ∭_V 3 dV = 3 Volume(sphere) = 3 ((4/3) π R^3) = 4π R^3.
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