lecture notes 13

.Example. Let S be the sphere x 2 + ∫∫y 2 + z 2 = 4, which bounds a solidball D. Calculate the outward flux F · n dS, whereF(x, . y, z) = (x 2 + y 2 + z 2 )(xi + yj + zk).Solution. Note that the divergence ∇ · F(x, y, z)= (x 3 + xy 2 + xz 2 ) x + (x 2 y + y 3 + yz 2 ) y + (x 2 z + y 2 z + z 3 ) z= 5(x 2 + y 2 + z 2 ). It follows from the divergence theorem and thenuse ∫∫ spherical coordinates ∫∫∫ that ∫∫∫F · n dS = ∇ · F dV = 5(x 2 + y 2 + z 2 ) dV==S∫ 2π ∫ π ∫ a∫0a0005ρ 4 dρ ×D5ρ 2 · ρ 2 sin ϕ dρ dϕ dθ∫ π0sin ϕ dϕ ×∫ 2π= a 5 × [1 − (−1)] × 2π = 4πa 5 .0DdθS. . . . . .