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...

In physics, the Young–Laplace equation (/ləˈplɑːs/) is an algebraic equation that describes the capillary pressure difference sustained across the interface...

For fluids in general, the speed of sound c is given by the Newton–Laplace equation: c = K s ρ , {\displaystyle c={\sqrt {\frac {K_{s}}{\rho }}},} where...

of probability was developed mainly by Laplace. Laplace formulated Laplace's equation, and pioneered the Laplace transform which appears in many branches...

(force) field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after French mathematician and...

Grattan-Guinness, I (1997), "Laplace's integral solutions to partial differential equations", in Gillispie, C. C. (ed.), Pierre Simon Laplace 1749–1827: A Life in...

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...

harmonics originate from solving Laplace's equation in the spherical domains. Functions that are solutions to Laplace's equation are called harmonics. Despite...

field v satisfies Laplace's equation. However, the converse is not true; not every vector field that satisfies Laplace's equation is a Laplacian vector...

wave equation Burgers' equation Continuity equation Heat equation Helmholtz equation Klein–Gordon equation Jacobi equation Lagrange equation Laplace's equation...

states that every weak solution of Laplace's equation is a smooth solution. This contrasts with the wave equation, for example, which has weak solutions...

Laplace's law or The law of Laplace may refer to several concepts, Biot–Savart law, in electromagnetics, it describes the magnetic field set up by a steady...

method of separation of variables for partial differential equations for Laplace's equation in polar form. This means that you write u ( r , θ ) = R (...

above properties mean, more precisely, that Laplace's equation for u {\displaystyle u} and Laplace's equation for v {\displaystyle v} are the integrability...

subset of ⁠ R n , {\displaystyle \mathbb {R} ^{n},} ⁠ that satisfies Laplace's equation, that is, ∂ 2 f ∂ x 1 2 + ∂ 2 f ∂ x 2 2 + ⋯ + ∂ 2 f ∂ x n 2 = 0 {\displaystyle...

the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the elliptic partial differential equation: ∇ 2 f = − k 2...

tides is described by the horizontal structure equation which is also called Laplace's tidal equation: L Θ n + ε n Θ n = 0 {\displaystyle {L}{\Theta }_{n}+\varepsilon...

Legendre's differential equation arises naturally whenever one solves Laplace's equation (and related partial differential equations) by separation of variables...

relationship is a form of Poisson's equation. In the absence of unpaired electric charge, the equation becomes Laplace's equation: ∇ 2 ϕ = 0 , {\displaystyle...

equation Heat equation Laplace's equation Laplace operator Harmonic function Spherical harmonic Poisson integral formula Klein–Gordon equation Korteweg–de...

Helmholtz equation in spherical coordinates. Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical...

differential equation is frequently encountered in physics and other technical fields. In particular, it occurs when solving Laplace's equation (and related...

PDEs. For example, a time-independent solution of the heat equation solves Laplace's equation. That is, if parabolic and hyperbolic PDEs are associated...

applications, such as when solving Laplace's equation in polar coordinates. The second order Cauchy–Euler equation is x 2 d 2 y d x 2 + a x d y d x +...

continuity equation ∇ ⋅ v → = 0 {\displaystyle \nabla \cdot {\vec {v}}=0} , the scalar potential can be substituted back in to find Laplace's Equation for irrotational...

function. One of the most well-known of these, the Laplace expansion for the three-variable Laplace equation, is given in terms of the generating function...

sloshing in tanks. If a function is harmonic (that is, it satisfies Laplace's equation ∇ 2 f = 0 {\displaystyle \nabla ^{2}f=0} ) over a plane domain (which...

Poisson's equation—or in the vacuum, Laplace's equation. There is considerable overlap between potential theory and the theory of Poisson's equation to the...

differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important...

known potentials, the potential away from them is obtained by solving Laplace's equation, either analytically or numerically. The electric field is then calculated...