The Levi-Civita symbol is a mathematical object used in vector calculus and tensor analysis, particularly in the context of multivariable calculus and differential geometry. It is denoted by ε and is defined for n-dimensional spaces, where n is typically 2 or 3. The symbol takes values of +1, -1, or 0, depending on the orientation of the vectors involved. In three dimensions, the Levi-Civita symbol is often represented as ε_ijk, where i, j, and k are indices that can take values 1, 2, or 3. It is used to express the cross product of vectors and to compute the determinant of matrices, making it essential in physics and engineering applications.