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A second-tensor rank symmetric tensor is defined as a tensor A for which A^(mn)=A^(nm). (1) Any tensor can be written as a sum of symmetric and antisymmetric parts A^(mn) = ...

The hyperbolic functions sinhz, coshz, tanhz, cschz, sechz, cothz (hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cosecant, hyperbolic secant, and ...

The permutation symbol (Evett 1966; Goldstein 1980, p. 172; Aris 1989, p. 16) is a three-index object sometimes called the Levi-Civita symbol (Weinberg 1972, p. 38; Misner et ...

A contravariant tensor is a tensor having specific transformation properties (cf., a covariant tensor). To examine the transformation properties of a contravariant tensor, ...

A covariant tensor, denoted with a lowered index (e.g., a_mu) is a tensor having specific transformation properties. In general, these transformation properties differ from ...

Minkowski space is a four-dimensional space possessing a Minkowski metric, i.e., a metric tensor having the form dtau^2=-(dx^0)^2+(dx^1)^2+(dx^2)^2+(dx^3)^2. Alternatively ...

The concept of a space is an extremely general and important mathematical construct. Members of the space obey certain addition properties. Spaces which have been ...

Roughly speaking, the metric tensor g_(ij) is a function which tells how to compute the distance between any two points in a given space. Its components can be viewed as ...

The distance between two points is the length of the path connecting them. In the plane, the distance between points (x_1,y_1) and (x_2,y_2) is given by the Pythagorean ...

The Laplacian for a scalar function phi is a scalar differential operator defined by (1) where the h_i are the scale factors of the coordinate system (Weinberg 1972, p. 109; ...