The Levi-Civita symbol is a mathematical object used in various fields, including vector calculus and tensor analysis. It is denoted as ε and is defined for a given dimension, typically in three dimensions as ε_ijk. The symbol takes the value of +1 for even permutations of its indices, -1 for odd permutations, and 0 if any indices are repeated. This symbol is particularly useful in calculating the cross product of vectors and in expressing the determinant of matrices. It helps simplify complex expressions involving multilinear algebra and is essential in the study of differential geometry and theoretical physics.