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A Cartesian tensor is a tensor in three-dimensional Euclidean space. Unlike general tensors, there is no distinction between covariant and contravariant indices for Cartesian ...

There are at least two theorems known as Chebyshev's theorem. The first is Bertrand's postulate, proposed by Bertrand in 1845 and proved by Chebyshev using elementary methods ...

Given a symmetric positive definite matrix A, the Cholesky decomposition is an upper triangular matrix U with strictly positive diagonal entries such that A=U^(T)U. Cholesky ...

The first type of tensor-like object derived from a Riemannian metric g which is used to study the geometry of the metric. Christoffel symbols of the first kind are variously ...

A set of orthogonal functions {phi_n(x)} is termed complete in the closed interval x in [a,b] if, for every piecewise continuous function f(x) in the interval, the minimum ...

The computational paradigm is a term introduced by Wolfram (2002, 2021) to describe the idea of using simple programs rather than mathematical equations (the latter of which ...

The Condon-Shortley phase is the factor of (-1)^m that occurs in some definitions of the spherical harmonics (e.g., Arfken 1985, p. 682) to compensate for the lack of ...

The second-order ordinary differential equation xy^('')+(c-x)y^'-ay=0, sometimes also called Kummer's differential equation (Slater 1960, p. 2; Zwillinger 1997, p. 124). It ...

(x^2)/(a^2-lambda)+(y^2)/(b^2-lambda)=z-lambda (1) (x^2)/(a^2-mu)+(y^2)/(b^2-mu)=z-mu (2) (x^2)/(a^2-nu)+(y^2)/(b^2-nu)=z-nu, (3) where lambda in (-infty,b^2), mu in ...

There are several different definitions of conical coordinates defined by Morse and Feshbach (1953), Byerly (1959), Arfken (1970), and Moon and Spencer (1988). The ...