Value of n for rⁿ r solenoidal

A vector field F is solenoidal if ∇ · F = 0 everywhere (except possibly at the origin).

Let r = xi + yj + zk and r = |r| = √(x² + y² + z²), so ∂r/∂x = x/r, and so on.

Divergence of rⁿ r

Using ∇ · (φF) = ∇φ · F + φ(∇ · F) with φ = rⁿ and F = r:

∇(rⁿ) = n r^(n−1) ∇r = n r^(n−1)(r/r) = n r^(n−2) r
∇ · (rⁿ r) = n r^(n−2) (r · r) + rⁿ (∇ · r)
           = n r^(n−2) r² + 3 rⁿ
           = (n + 3) rⁿ

Condition

(n + 3) rⁿ = 0 for all r ≠ 0 requires n + 3 = 0.

Answer: n = −3. The field r/r³ (the inverse-square field) is solenoidal.