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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.
The left-hand side of this equation is the Laplace operator, and the entire equation Δu = 0 is known as Laplace's equation. Solutions of the Laplace equation, i.e. functions whose Laplacian is identically zero, thus represent possible equilibrium densities under diffusion.
is the fundamental solution of the Laplace equation in the upper half-plane. [59] It represents the electrostatic potential in a semi-infinite plate whose potential along the edge is held at fixed at the delta function.
In mathematics, the Laplace transform, named after Pierre-Simon Laplace (/ l ə ˈ p l ɑː s /), is an integral transform that converts a function of a real variable (usually , in the time domain) to a function of a complex variable (in the complex-valued frequency domain, also known as s-domain, or s-plane).
However, the converse is not true; not every vector field that satisfies Laplace's equation is a Laplacian vector field, which can be a point of confusion. For example, the vector field v = ( x y , y z , z x ) {\displaystyle {\bf {v}}=(xy,yz,zx)} satisfies Laplace's equation, but it has both nonzero divergence and nonzero curl and is not a ...
The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form:
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In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function:, where U is an open subset of , that satisfies Laplace's equation, that is, + + + = everywhere on U.