# What is Laplace equation: Definition and 155 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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1. ### I Question regarding Laplace's Equation for regions with charges

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2. ### I Laplace equation with irregular boundaries

Is there a way to solve Laplace’s Equation on irregular domains if the domain’s shape is given by a function for example a 2D parabolic plate. I keep seeing numerical methods but I want to know is there an ANALYTICAL method to solve it on an irregular domain. If there isn't are there approximate...
3. ### I Laplace's equation in presence of a dipole (perfect or physical)

Does Laplace's equation hold true for electrostatic potential at the location of a dipole? Or should poisson's equation be used?
4. ### I Where to find this uniqueness theorem of electrostatics?

There is a nice uniqueness theorem of electrostatics, which I have found only after googling hours, and deep inside some academic site, in the lecture notes of Dr Vadim Kaplunovsky: Notice that the important thing here is that only the NET charges on the conductors are specified, not their...
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46. ### Problem with solving laplace equation with a charged ring

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48. ### Magnetic Mirror for Neutral Atoms

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49. ### Solve Laplace equation on rectangle domain

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

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