Conormal derivative (${\partial }_{n,K}u=\mathbf{n}\cdot K\nabla u$)

For a divergence-form differential operator $-\nabla \cdot K\nabla$, its conormal derivative is the flux factor appearing in integration by parts. It generally differs from the ordinary normal derivative ${\partial }_{n}u=\mathbf{n}\cdot \nabla u$ when $K$ is anisotropic. Decomposing the gradient along a smooth boundary into its normal and tangential components gives ${\partial }_{n,K}u=(\mathbf{n}\cdot K\mathbf{n}){\partial }_{n}u+(\mathbf{n}\cdot K\mathbf{t}){\partial }_{\mathbf{t}}u$ in two dimensions. Thus zero normal derivative does not in general remove the boundary term in the usual symmetric weak formulation. A zero conormal derivative does remove it and produces the natural homogeneous Neumann boundary condition for that energy.