Apply the product rule and the preceding gradient formula to the vector field

F_{i} = f (r) x_{i}

. Summing over the repeated index gives

\partial_{i} F_{i} = f^{'} (r) \frac{x _{i} x _{i}}{r} + f (r) δ_{ii} = r f^{'} (r) + n f (r) .

(1)

Here

δ_{ii} = n

for the Kronecker delta. Thus the divergence and curl of a radial vector field give, in any dimension,

\nabla \cdot [f (r) x] = r f^{'} (r) + n f (r) .

(2)

Both terms contract an antisymmetric array with a symmetric one, and therefore vanish. Every differentiable radial field of this form has zero curl away from the origin. This is the three-dimensional part of the divergence and curl of a radial vector field.