LaTeX
\nabla^2u=0Variables
- u: scalar field
- ∇²: Laplacian operator
How to use this formula
Defines harmonic functions and steady-state potential fields.
Important notes
- Boundary conditions determine the solution.
- It is the steady-state form of the heat equation without sources.
Quick example
Electrostatic potential in a charge-free region satisfies ∇²u=0.
Applicability, worked calculation, and verification
Assumptions and domain checks
- Boundary conditions determine the solution.
- State the domain together with any initial or boundary conditions, and confirm the regularity assumptions required by the method.
Worked example
Electrostatic potential in a charge-free region satisfies ∇²u=0.
Common mistakes
- Before substituting values into Laplace Equation, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.
- Do not present a general solution as the requested solution before applying all initial or boundary conditions.
Continue the workflow
Use Laplace Equation in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
\nabla^2u=0. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Digital Library of Mathematical FunctionsNational Institute of Standards and Technology — Definitions, notation, identities, and reference material for mathematical functions.
Frequently asked questions
What is the Laplace Equation used for?
Defines harmonic functions and steady-state potential fields.
Can I copy this formula as LaTeX?
Yes. Copy \nabla^2u=0 or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.