# P-Vector

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## 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*, and

_{ji}*b*is a contravariant 3-vector if

^{jki}*b*= —

^{ijk}*b*=

^{jik}*b*= —

^{jki}*b*=

^{ikj}*b*= —

^{kij}. If only those comonents of the *m*-vector

*ω*_{i1}, *i*_{2}, …, *i _{m}*

are retained for which *i*_{1} < *i*_{2} < … < *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.