Surface tangent projector ($P=I-\mathbf{nn}$)

The orthogonal projector onto the tangent plane of a surface with unit normal vector $\mathbf{n}$. It removes the normal component of a vector and defines the surface gradient ${\nabla }_s=P\nabla$.

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.