Laplace's equation in the context of "Hyperbolic functions"

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.

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 ⁠${\displaystyle \nabla \cdot \nabla }$⁠, ${\displaystyle \nabla ^{2}}$ (where ${\displaystyle \nabla }$ is the nabla operator), or ⁠${\displaystyle \Delta }$⁠. 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 Δf (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 Δf = 0 are called harmonic functions and represent the possible gravitational potentials in regions of vacuum.

In mathematics and mathematical physics, potential theory is the study of harmonic functions.

The term "potential theory" dates from 19th-century physics when it was realized that the 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 satisfy Poisson's equation—or in the vacuum, Laplace's equation.