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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, aij is a covariant 2-vector if aij = — aji, and bjki is a contravariant 3-vector if bijk = —bjik = bjki = —bikj = bkij = —
. If only those comonents of the m-vector
ωi1, i2, …, im
are retained for which i1 < i2 < … < im, 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.