Laplace's equation

Laplace's_equation_on_an_annulus

Rotating_spherical_harmonics

Laplace's equation is a second-order partial differential equation named after the French mathematician Pierre-Simon Laplace. It is a fundamental equation in the field of potential theory and appears in many areas of physics and engineering, particularly in the study of electrostatics, gravitation, and fluid dynamics.

Mathematical Formulation

Laplace's equation is given by: \[ \Delta \phi = 0 \] where \( \Delta \) is the Laplace operator (or Laplacian) and \( \phi \) is a twice-differentiable function. In three-dimensional Cartesian coordinates, the Laplace operator is expressed as: \[ \Delta \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2} \]

Applications

Laplace's equation is widely used in various fields:

• In electrostatics, it describes the potential field generated by a distribution of electric charges in the absence of free charges.
• In fluid dynamics, it is used to describe the velocity potential of an incompressible and irrotational fluid flow.
• In gravitation, it describes the gravitational potential in a region with no mass.

Boundary Conditions

Solutions to Laplace's equation are determined by the boundary conditions of the problem. Common types of boundary conditions include:

• Dirichlet boundary condition: Specifies the value of the function \( \phi \) on the boundary.
• Neumann boundary condition: Specifies the value of the normal derivative of \( \phi \) on the boundary.
• Robin boundary condition: A combination of Dirichlet and Neumann boundary conditions.

Harmonic Functions

A function that satisfies Laplace's equation is called a harmonic function. Harmonic functions have several important properties:

• They are infinitely differentiable within the domain.
• They satisfy the mean value property.
• They exhibit the maximum principle, meaning that the maximum and minimum values of a harmonic function occur on the boundary of the domain.

Related Equations

Laplace's equation is a special case of the more general Poisson's equation, which is given by: \[ \Delta \phi = f \] where \( f \) is a known function. When \( f = 0 \), Poisson's equation reduces to Laplace's equation.

See Also

• Poisson's equation
• Harmonic function
• Laplace operator
• Dirichlet problem
• Neumann problem
• Potential theory