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Canonical static concept record

Laplace operator

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Summary

The divergence of the gradient, or its geometric Laplace–Beltrami generalization.

Record metadata

Canonical name: Laplace operator
Concept ID: laplace_operator
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Analysis
Domains: Analysis
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Differential operator

Carrier(s)

Riemannian manifold or Euclidean domain

Data

second-order operator $\Delta$

Axioms / constraints

defined from the metric/divergence-gradient convention

Canonically induces

harmonic functions
spectral decomposition under suitable boundary conditions

Notes

The overall sign convention varies.

Concept sources

Wikipedia — Laplace operator — Laplace operator · encyclopedic reference · source ID wp-laplace_operator
nLab — Laplace operator — Laplace operator · mathematical reference · source ID nlab-laplace_operator
Cambridge — Quantum Mechanics — Quantum Mechanics · University of Cambridge lecture notes · source ID tong-quantum

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Differential operator Laplace operator

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: laplace_from_differential
Relation type: Framework specialization framework-specialization
Direction: source → target
Endpoint roles: source: Specializes to; target: Special case or refinement of
Authored annotation: second-order Laplacian

Authored explanation

The Laplace or Laplace–Beltrami operator is a second-order differential operator.

How to interpret this relation type

A more specific theory, model, entity class, or regime is obtained by restricting or extending the scope of a broader framework.

Relation sources

Wikipedia — Differential operator — Differential operator · encyclopedic reference · source ID wp-differential_operator
nLab — differential operator — differential operator · mathematical reference · source ID nlab-differential_operator
Cambridge — Quantum Field Theory — Quantum Field Theory · University of Cambridge lecture notes · source ID tong-qft
Wikipedia — Laplace operator — Laplace operator · encyclopedic reference · source ID wp-laplace_operator
nLab — Laplace operator — Laplace operator · mathematical reference · source ID nlab-laplace_operator
Cambridge — Quantum Mechanics — Quantum Mechanics · University of Cambridge lecture notes · source ID tong-quantum

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.