Skip to main content

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

Short answer

Divergence is 3. Volume of unit sphere is 4π/3, so flux = 3 × 4π/3 = 4π.

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

Statement

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

∬_S F·n dS = ∭_V (∇·F) dV.

Evaluation

F = x i + y j + z k
div F = 1 + 1 + 1 = 3
Volume of unit sphere = 4π/3

∬_S F·n dS = ∭_V 3 dV = 3 × 4π/3 = 4π

Check: on the unit sphere n = x i + y j + z k, so F·n = x² + y² + z² = 1 and the integral is the area of the sphere, 4π(1)² = 4π.

Answer: 4π.

✓ Answer checked by Claude Opus 5.5 (high effort)

Related questions

All Mathematics III chapter 2 questions →