Sphere surface derivative identities
= Sphere surface derivative identities
{title2=$\Delta_s\mathbf n=-2\mathbf n/a^2$}
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.