Div and curl of grad φ, φ=log(x²+y²+z²)

φ = log(x² + y² + z²). Let s = x² + y² + z².

Gradient

∂φ/∂x = 2x/s,   ∂φ/∂y = 2y/s,   ∂φ/∂z = 2z/s
grad φ = (2/s)(x i + y j + z k) = 2r/s

Divergence of grad φ

∂/∂x (2x/s) = 2/s − 2x(2x)/s² = 2/s − 4x²/s²

Similarly for y and z. Add:

div grad φ = 6/s − 4(x² + y² + z²)/s²
           = 6/s − 4/s = 2/s

div(grad φ) = 2/(x² + y² + z²).

Curl of grad φ

grad φ = (2x/s, 2y/s, 2z/s). The x-component of the curl is

∂/∂y (2z/s) − ∂/∂z (2y/s)
= 2z(−2y/s²) − 2y(−2z/s²) = −4yz/s² + 4yz/s² = 0

The other components vanish in the same way (this is the identity curl grad φ = 0 for any twice-differentiable φ).

curl(grad φ) = 0.