# What is Laplace's equation: Definition and 116 Discussions

In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as

2

f
=
0

or

Δ
f
=
0
,

where

Δ
=

=

2

{\displaystyle \Delta =\nabla \cdot \nabla =\nabla ^{2}}
is the Laplace operator,

{\displaystyle \nabla \cdot }
is the divergence operator (also symbolized "div"),

{\displaystyle \nabla }

f
(
x
,
y
,
z
)

{\displaystyle f(x,y,z)}
is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function.
If the right-hand side is specified as a given function,

h
(
x
,
y
,
z
)

{\displaystyle h(x,y,z)}
, we have

Δ
f
=
h
.

{\displaystyle \Delta f=h.}
This is called Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations. Laplace's equation is also a special case of the Helmholtz equation.
The general theory of solutions to Laplace's equation is known as potential theory. The solutions of Laplace's equation are the harmonic functions, which are important in multiple branches of physics, notably electrostatics, gravitation, and fluid dynamics. In the study of heat conduction, the Laplace equation is the steady-state heat equation. In general, Laplace's equation describes situations of equilibrium, or those that do not depend explicitly on time.

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2. ### I Question regarding Laplace's Equation for regions with charges

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3. ### Laplace's equation when there is a point dipole

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4. ### I Laplace's equation in presence of a dipole (perfect or physical)

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5. ### A Solving Laplace's equation in polar coordinates for specific boundary conditions

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6. ### I Unraveling Laplace's Equation: Exploring Valid Domains and Boundary Conditions

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7. ### Thin disc with high thermal conductivity

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10. ### I PDEs: Laplace's Equation over a Parallelogram

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11. ### Laplace's equation vs Principle of Least Action?

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13. ### I Solve Laplace's Equation with Laplace Transform

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15. ### I Can the Schrodinger equation satisfy Laplace's equation?

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17. ### Finite element solving of Laplace's equation doesn't converge

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18. D

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19. ### Solution to Laplace's Equation

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20. ### 2D Laplace's equation with mixed boundary conditions

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21. ### Potential of a Rectangular Pipe by Laplace's Equation

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22. ### Solution to Laplace's equation in parabolic coordinates

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26. ### Use of Laplace's equation in analytic function theory

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28. ### RHS of Laplace's Equation is f(u(x,y))

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