Conormal derivative
= Conormal derivative
{title2=$\partial_{n,K}u=\mathbf n\cdot K\nabla u$}
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> $\partial_nu=\mathbf n\cdot\nabla u$ when $K$ is anisotropic. Decomposing the <gradient> along a smooth boundary into its normal and tangential components gives $\partial_{n,K}u=(\mathbf n\cdot K\mathbf n)\partial_nu+(\mathbf n\cdot K\mathbf t)\partial_{\mathbf t}u$ 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.