Question

Given a function φ = ½(x²+y²+z²) in three-dimensional Cartesian space, the value of the surface integral ∮∮_S n̂·∇φ dS, where S is the surface of a sphere of unit radius and n̂ is the outward unit normal vector on S, is

Accepted Answer

The correct answer is (A) 4π. ∇φ = (x,y,z), the position vector. On the unit sphere, n̂ equals the position vector itself, so n̂·∇φ = |r|² = 1 everywhere on S. The integral is then just the surface area of the unit sphere: 4π(1)² = 4π. (Cross-check via the divergence theorem: ∇²φ=3, so the integral = ∫∫∫3 dV = 3×(4π/3) = 4π.)

Explanation