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⭐ Core Definition: Laplace operator

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols ⁠ $\nabla \cdot \nabla$ ⁠, $\nabla ^{2}$ (where $\nabla$ is the nabla operator), or ⁠ $\Delta$ ⁠. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian $Δ f (p)$ of a function $f$ at a point $p$ measures by how much the average value of $f$ over small spheres or balls centered at $p$ deviates from $f (p)$ .

The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics: the Laplacian of the gravitational potential due to a given mass density distribution is a constant multiple of that density distribution. Solutions of Laplace's equation $Δ f = 0$ are called harmonic functions and represent the possible gravitational potentials in regions of vacuum.

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⭐ Core Definition: Laplace operator
Laplace operator in the context of Pierre-Simon Laplace
Laplace operator in the context of Laplace's equation

Laplace operator in the context of Pierre-Simon Laplace

Pierre-Simon, Marquis de Laplace (/ləˈplɑːs/; French: [pjɛʁ simɔ̃ laplas]; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summarized and extended the work of his predecessors in his five-volume Mécanique céleste (Celestial Mechanics) (1799–1825). This work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. Laplace also popularized and further confirmed Sir Isaac Newton's work. In statistics, the Bayesian interpretation of probability was developed mainly by Laplace.

Laplace formulated Laplace's equation, and pioneered the Laplace transform which appears in many branches of mathematical physics, a field that he took a leading role in forming. The Laplacian differential operator, widely used in mathematics, is also named after him. He restated and developed the nebular hypothesis of the origin of the Solar System and was one of the first scientists to suggest an idea similar to that of a black hole, with Stephen Hawking stating that "Laplace essentially predicted the existence of black holes". He originated Laplace's demon, which is a hypothetical all-predicting intellect. He also refined Newton's calculation of the speed of sound to derive a more accurate measurement.

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Laplace operator in the context of Laplace's equation

In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties in 1786. This is often written as $\nabla ^{2}\!f=0$ or $\Delta f=0,$ where $\Delta =\nabla \cdot \nabla =\nabla ^{2}$ is the Laplace operator, $\nabla \cdot$ is the divergence operator (also symbolized "div"), $\nabla$ is the gradient operator (also symbolized "grad"), and $f(x,y,z)$ is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function.