{title2=$\partial_N$}
{wiki=Directional_derivative}

The normal derivative of a differentiable function $u$ along a hypersurface with <unit normal> $N$ is the <directional derivative>
$$
\partial_Nu=N\mathbin\cdot\nabla u.
$$
Higher normal derivatives at a point $x$ may be defined by differentiating the one-variable function $s\mapsto u(x+sN(x))$ repeatedly at $s=0$.


Normal derivative

{title2=$D_vf$}
{wiki}

The directional derivative of $f$ at $x$ in the direction $v$ is
$$
D_vf(x)=\left.\frac d{dt}f(x+tv)\right|_{t=0}.
$$
For a differentiable scalar function on Euclidean space, it equals $v\mathbin\cdot\nabla f(x)$.


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 Incoming links: Directional derivative