[1] When imposed on an ordinary or a partial differential equation, the condition specifies the values of the derivative applied at the boundary of the domain.
For an ordinary differential equation, for instance, the Neumann boundary conditions on the interval [a,b] take the form where α and β are given numbers.
For a partial differential equation, for instance, where ∇2 denotes the Laplace operator, the Neumann boundary conditions on a domain Ω ⊂ Rn take the form where n denotes the (typically exterior) normal to the boundary ∂Ω, and f is a given scalar function.
The normal derivative, which shows up on the left side, is defined as where ∇y(x) represents the gradient vector of y(x), n̂ is the unit normal, and ⋅ represents the inner product operator.
It becomes clear that the boundary must be sufficiently smooth such that the normal derivative can exist, since, for example, at corner points on the boundary the normal vector is not well defined.