P-Vector

a tensor that is skew-symmetric with respect to any two of its indices. It thus is a tensor with either only covariant indices (subscripts) or only contravariant indices (superscripts), where each index can take on values from 1 to n. Moreover, a component of a p-vector changes sign when any two of the component’s indices are interchanged.

If the degree—that is, the number of indices—of a p-vector is equal to 2, 3…. m, we speak of a 2-vector, 3-vector, …, m-vector, respectively. For example, a_ij is a covariant 2-vector if a_ij = — a_ji, and b^jki is a contravariant 3-vector if b^ijk = —b^jik = b^jki = —b^ikj = b^kij = —

. If only those comonents of the m-vector

ω_i1, i₂, …, i_m

are retained for which i₁ < i₂ < … < i_m, essential components will remain.

The components of a p-vector can be arranged in a certain way in the form of a rectangular matrix of n rows and columns whose rank is called the rank of the p-vector. If a p-vector’s rank is equal to its degree (valency), the p-vector is the exterior product of tensors of degree one and is said to be simple.