Sphere surface derivative identities (${\Delta }_s\mathbf{n}=-2\mathbf{n}/a^2$)

On a sphere of radius $a$, ${\nabla }_s\mathbf{n}=P/a$, ${\nabla }_s\cdot \mathbf{n}=2/a$, and ${\Delta }_s\mathbf{n}=-2\mathbf{n}/a^2$. For a constant vector $\mathbf{U}$, ${\nabla }_s\cdot (P\mathbf{U})=-2\mathbf{U}\cdot \mathbf{n}/a$. These identities simplify dipolar surface diffusion and interfacial transport.