Signature

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Values at fixed points

General characteristics

Domain and analyticity

(Subscript[ε, (Subscript[n, 1]), (Subscript[n, 2]),… , (Subscript[n, d])]) is a nonanalytical function, defined on the set of tuples of integers with possible values 0,±1.

Symmetries and periodicities

Quasi-permutation symmetry

Transformations

Products, sums, and powers of the direct function

Products of the direct function

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Summation

Finite summation

Theorems

The determinant of a matrix with entries elements

The determinant det(A) of the n×n matrix A with entries (Subscript[a, i j]) can be expressed as

det(A)==(∑_((Subscript[k, 1])=1)^n)(∑_((Subscript[k, 2])=1)^n)…(∑_((Subscript[k, n])=1)^n)(Subscript[ε, (Subscript[k, 1])(Subscript[k, 2]),…,(Subscript[k, n])])(Subscript[a, 1(Subscript[k, 1])])(Subscript[a, 2(Subscript[k, 2])])… (Subscript[a, n (Subscript[k, n])]).

Antisymmetric Levi-Civita tensor

The set of all  (Subscript[ε, (Subscript[n, 1]),(Subscript[n, 2]),…,(Subscript[n, d])]) /;1≤(Subscript[n, 1])≤d,…, 1≤(Subscript[n, d])≤d  forms the completly antisymmetric Levi-Civita tensor of dimension d.

History

– T. Levi–Civita (1896)

Copyright