Is the Laplace equation the same as Laplace's equation?

Yes, 'Laplace equation' and 'Laplace's equation' are used interchangeably, though the latter is slightly more possessive in style.

What is the main physical interpretation of the Laplace equation?

It describes equilibrium or steady-state phenomena, such as the temperature distribution in a stationary object or the electric potential in a region without charge.

Can the Laplace equation have non-zero solutions?

Yes, absolutely. The equation ∇²ψ = 0 has many non-trivial (non-zero) solutions, which are determined by the boundary conditions of the problem.

What is the difference between the Laplace and Poisson equations?

The Laplace equation is ∇²ψ = 0. The Poisson equation is ∇²ψ = f (a source term). So Laplace is a special case of Poisson where the source term f is zero.

laplace equation

UK/lɑːˈplɑːs ɪˌkweɪʒən/US/lɑˈplɑs ɪˈkweɪʒən/

Academic/Technical

My Flashcards

Definition

Meaning

A second-order partial differential equation, denoted as ∇²ψ = 0, which describes a function whose value at any point is the average of its values in an infinitesimal neighborhood.

In mathematics and physics, a fundamental equation describing harmonic functions, which appear in electrostatics, fluid dynamics, gravitation, and steady-state heat conduction.

Linguistics

Semantic Notes

Named after Pierre-Simon Laplace. Always capitalized as it is a proper noun derivative. Used primarily as a singular noun phrase ('the Laplace equation'), though sometimes referred to as 'Laplace's equation'.

Dialectal Variation

British vs American Usage

Differences

No significant lexical differences. Spelling of related terms (e.g., 'behaviour/behavior', 'centre/center') follows regional conventions.

Connotations

Identical technical meaning. Pronunciation of 'Laplace' may slightly differ (see IPA).

Frequency

Equally frequent in relevant academic/technical contexts in both varieties.

Vocabulary

Collocations

strong

solve the Laplace equationsatisfy the Laplace equationLaplace equation in spherical coordinatesboundary conditions for the Laplace equation

medium

numerical solution of the Laplace equationapply the Laplace equationderive the Laplace equation

weak

classical Laplace equationcomplex Laplace equationfundamental Laplace equation

Grammar

Valency Patterns

The Laplace equation governs [physical phenomenon][Function/Field] satisfies the Laplace equationA solution to the Laplace equation is called harmonic

Vocabulary

Synonyms

Strong

harmonic equation

Neutral

Laplace's equation

Weak

potential equation

Vocabulary

Antonyms

Poisson equationinhomogeneous Laplace equation

Phrases

Idioms & Phrases

“It's not Laplace's equation (informal metaphor for something simple or elegantly balanced)”

Usage

Context Usage

Business

Virtually never used.

Academic

Core concept in mathematics, physics, and engineering courses; used in research papers and textbooks.

Everyday

Never used in everyday conversation.

Technical

Fundamental in fields like electromagnetism, fluid mechanics, and mathematical physics for modeling potential fields.

Examples

By Part of Speech

verb

British English

The potential field laplaces to zero in the source-free region.
One must laplace the differential operator to check for harmonicity.

American English

The function laplaces across the domain, indicating equilibrium.
You can laplace that expression to see if it's harmonic.

adverb

British English

The field varied Laplace-equation-like throughout the volume.
The system behaved almost Laplace-equation-ically.

American English

The distribution spread Laplace-equation-style.
It decayed, but not Laplace-equation-fast.

adjective

British English

The Laplace-equation solution is harmonic.
We studied the Laplace-equation properties.

American English

The Laplace-equation form is elliptic.
A Laplace-equation solver is needed.

Examples

By CEFR Level

(Not applicable at this level)

(Not applicable at this level)

The Laplace equation is a key topic in university physics.
Engineers use the Laplace equation to model heat flow.

Solving the Laplace equation with complex boundary conditions requires advanced numerical methods.
The behaviour of the electrostatic potential in a charge-free region is governed by the Laplace equation.

Learning

Memory Aids

Mnemonic

Imagine a perfectly flat, still pond (zero curvature). Laplace's equation describes such a state of perfect balance for a mathematical surface.

Conceptual Metaphor

Mathematical harmony or balance; a state of equilibrium where every point is the average of its surroundings.

Watch out

Common Pitfalls

Translation Traps (for Russian speakers)

Avoid translating as 'уравнение Лапласа' in a non-technical context where it would be meaningless.
Do not confuse with 'Laplace transform' (преобразование Лапласа), which is a different concept.

Common Mistakes

Misspelling as 'La Place equation'.
Using lowercase 'l' (it's a proper name).
Confusing it with the 'Laplace operator' (∇²), which is the operator within the equation.

Practice

Quiz

Fill in the gap

In electrostatics, the electric potential in a region with no charge density must satisfy the .

Multiple Choice

What is a defining property of a function that solves the Laplace equation?

FAQ

Frequently Asked Questions

: Yes, 'Laplace equation' and 'Laplace's equation' are used interchangeably, though the latter is slightly more possessive in style.
: It describes equilibrium or steady-state phenomena, such as the temperature distribution in a stationary object or the electric potential in a region without charge.
: Yes, absolutely. The equation ∇²ψ = 0 has many non-trivial (non-zero) solutions, which are determined by the boundary conditions of the problem.
: The Laplace equation is ∇²ψ = 0. The Poisson equation is ∇²ψ = f (a source term). So Laplace is a special case of Poisson where the source term f is zero.