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Evaluate using Gauss' divergence theorem F=2xi+3yj+4zk

Short answer

Divergence is 9. Volume integral over unit sphere: 9 × (4π/3) = 12π.

State Gauss divergence theorem. Evaluate ∬_S F·n ds for F = 2x i + 3y j + 4z k where S is the surface of the sphere x² + y² + z² = 1 by Gauss's divergence theorem.

Statement

If S is a closed surface enclosing a volume V, n is the outward unit normal and F is a vector function with continuous first partial derivatives in V, then

∬_S F·n dS = ∭_V (∇·F) dV.

Evaluation

F = 2x i + 3y j + 4z k
div F = 2 + 3 + 4 = 9

S is the unit sphere, so V is the unit ball with volume (4/3)π(1)³ = 4π/3.

∬_S F·n dS = ∭_V 9 dV = 9 × (4π/3) = 12π

Answer: ∬_S F·n dS = 12π.

✓ Answer checked by Claude Opus 5.5 (high effort)

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