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

Short answer

Divergence ∇·F = 9 is constant. Volume of unit sphere is 4π/3, so flux = 9 × 4π/3 = 12π.

Using 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.

F = 2x i + 3y j + 4z k, S: the unit sphere x² + y² + z² = 1.

Apply Gauss' divergence theorem

∬_S F·n dS = ∭_V (∇·F) dV
∇·F = ∂(2x)/∂x + ∂(3y)/∂y + ∂(4z)/∂z = 2 + 3 + 4 = 9

The divergence is constant, so the integral is 9 times the volume of the unit ball, V = (4/3)π(1)³ = 4π/3:

∬_S F·n dS = 9 · (4π/3) = 12π

Answer: 12π.

(Check: on the unit sphere n = (x, y, z), so F·n = 2x² + 3y² + 4z². By symmetry the average of each of x², y², z² over the sphere is 1/3, giving (2 + 3 + 4)/3 · 4π = 12π ✓.)

✓ Answer checked by Claude Opus 5.5 (high effort)

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