# What would be an example of Neumann boundary conditions on a two dimensional domain

The Wikipedia page says that it would involve the derivative with respect to some normal vector being constant, but I don't quite understand this.

• Is the value of the normal vector adjusted across the domain, or does it remain the same?
• Can I say that the standard partial derivative is the derivative with respect to the normal vector if I have a rectangular domain?

Consider a domain (nonempty connected open set) $\Omega\subset{\bf R}^n$ with smooth boundary $\partial \Omega$. At each point $x\in\partial \Omega$, we have a normal vector $n(x)$.
Consider for instance $\Omega$ being a closed unit ball in ${\bf R}^n$ (in the case when $n=2$, you have a closed unit disk). One has $$n(x)=x.$$
A particular Neumann boundary condition looks like $$\frac{\partial u}{\partial n}(x)=f(x),\quad x\in\partial \Omega$$ which is understood as $$\nabla u(x)\cdot n(x)=f(x),\quad x\in\partial\Omega.$$