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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 = or =, where = = is the Laplace operator, [note 1] is the divergence operator (also symbolized "div"), is the gradient operator (also symbolized "grad"), and (,,) is a twice-differentiable real-valued function.
In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the derivative of the Green's function for the Laplace equation.
The Laplace operator is a second-order differential operator in the n-dimensional Euclidean space, defined as the divergence of the gradient (). Thus if f {\displaystyle f} is a twice-differentiable real-valued function , then the Laplacian of f {\displaystyle f} is the real-valued function defined by:
The two-dimensional analogue of the vibrating string is the vibrating membrane, with the edges clamped to be motionless. The Helmholtz equation was solved for many basic shapes in the 19th century: the rectangular membrane by Siméon Denis Poisson in 1829, the equilateral triangle by Gabriel Lamé in 1852, and the circular membrane by Alfred Clebsch in 1862.
In mathematics and mathematical physics, potential theory is the study of harmonic functions.. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravity and the electrostatic force, could be modeled using functions called the gravitational potential and electrostatic potential, both of which ...
The free-space circular cylindrical Green's function (see below) is given in terms of the reciprocal distance between two points. The expression is derived in Jackson's Classical Electrodynamics. [1] Using the Green's function for the three-variable Laplace operator, one can integrate the Poisson equation in
where the Laplace operator, Δ or ∇ 2, the divergence of the gradient, is taken in the spatial variables. The heat equation governs heat diffusion, as well as other diffusive processes, such as particle diffusion or the propagation of action potential in nerve cells. Although they are not diffusive in nature, some quantum mechanics problems ...
Considering higher dimensional analogues of the harmonics on the unit n-sphere, one arrives at the spherical harmonics. These functions satisfy Laplace's equation and over time "harmonic" was used to refer to all functions satisfying Laplace's equation.