In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre Simon de Laplace in 1782.
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Nabla symbol
∇ The nabla symbol Nabla is the symbol (∇). The name comes from the Greek word for a Hebrew harp, which had a similar shape. Related words also exist in Aramaic and Hebrew. The symbol was first used by William Rowan Hamilton in the form of a sideways wedge: ⊲. Another, less-common name for the symbol is atled (delta spelled backwards), because the nabla is an inverted Greek letter delta. In actual Greek usage, the symbol is called ανάδελτα, anádelta, which means "upside-down delta".
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Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste (1799�). This work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. In statistics, the so-called Bayesian interpretation of probability was mainly developed by Laplace.
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Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another (in the style of a higher-order function in computer science). This article considers mainly linear operators, which are the most common type. However, non-linear differential operators, such as the Schwarzian derivative also exist.
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Legendre transformation
is the value of the Legendre transform, where . Note that for any other point on the red curve, a line drawn through that point with the same slope as the blue line will have a y-intercept above the point, showing that is indeed a maximum. ]] In mathematics, the Legendre transformation or Legendre transform, named after Adrien-Marie Legendre, is an operation that transforms one real-valued function of a real variable into another.
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Helmholtz equation
The Helmholtz equation, named for Hermann von Helmholtz, is the partial differential equation where ∇ is the Laplacian, k is the wavenumber, and A is the amplitude.
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Legendre function
In mathematics, the Legendre functions Pλ, Qλ and associated Legendre functions Pμλ, Qμλ are generalizations of Legendre polynomials to non-integer degree.
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Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics, and other disciplines.
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