Editing Neumann boundary condition (section)

===PDE===

For a partial differential equation, for instance,

:<math>\nabla^2 y + y = 0,</math>

where {{math|∇<sup>2</sup>}} denotes the [[Laplace operator]], the Neumann boundary conditions on a domain {{math|Ω ⊂ '''R'''<sup>''n''</sup>}} take the form

:<math>\frac{\partial y}{\partial \mathbf{n}}(\mathbf{x}) = f(\mathbf{x}) \quad \forall \mathbf{x} \in \partial \Omega,</math>

where {{math|'''n'''}} denotes the (typically exterior) [[normal vector|normal]] to the [[boundary (topology)|boundary]] {{math|∂Ω}}, and {{mvar|f}} is a given [[scalar function]].

The [[normal derivative]], which shows up on the left side, is defined as

:<math>\frac{\partial y}{\partial \mathbf{n}}(\mathbf{x}) = \nabla y(\mathbf{x}) \cdot \mathbf{\hat{n}}(\mathbf{x}),</math>

where {{math|∇''y''('''x''')}} represents the [[gradient]] vector of {{math|''y''('''x''')}}, {{math|'''n̂'''}} is the unit normal, and {{math|⋅}} 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.