DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09

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Classical normal derivative

Definition

For a bounded $C^{1}$ domain $Ω$ and $u \in C^{1} (\overline{Ω})$ , the classical normal derivative is $\partial_{ν} u (x) = D u (x) \cdot ν (x)$ on $\partial Ω$ . Here Du is the continuous extension of the interior gradient and $ν$ is the outward unit normal. For a specified finite piecewise $C^{1}$ presentation define the same expression on each face off its edge set E; arbitrary values on E do not change a surface integral. This definition uses a classical continuous trace, and asserts neither a Sobolev trace nor a conormal derivative.

Source notes

Hunter §2.5, immediately before Theorem 2.23, printed p. 32 (PDF p. 38).

Depends on

Used by

Classical Neumann solutions differ by componentwise constants Corollary
First Green identity Corollary
Uniqueness of classical Dirichlet and compatible Neumann solutions Corollary
Neumann Poisson data require a flux compatibility equation Counterexample
Poisson kernel from a Dirichlet Green function Definition
Scalar conductivity and Schrödinger Dirichlet-to-Neumann maps Definition
Newtonian potential of radial compact data Example
Classical solutions satisfy the weak formulation Lemma
Green's second identity on a compact bordered domain of a Riemann surface Lemma
Heat-ball representation formula Lemma
Liouville transform and conductivity DN conjugacy Lemma
Necessary compatibility for the classical Neumann Poisson problem Lemma
Green and harmonic-measure representation with the 2π sign Theorem
Green representation for classical Poisson data Theorem
Symmetry of the Dirichlet Green function Theorem
The Gauss-Green integration-by-parts formula with Sobolev traces Theorem

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Hunter, Notes on Partial Differential Equations (standard reference, not scraped)