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Mathematics for Physics

171 k-Covector

An alternating multilinear mapping on k vectors that is used to span the set of alternating tensors of that degree.

definition (k-Covector) An alternating \(k\)-linear function from the \(k\)-fold product of a vector space to the real numbers, \(T: V^k \rightarrow \mathbb{R}\), where the following conditions apply:

  • The function \(T\) is linear in each of its \(k\) arguments.
  • \(T(v_{\sigma(1)}, \dots, v_{\sigma(k)}) = (\text{sgn } \sigma) T(v_1, \dots, v_k)\) for every permutation \(\sigma \in S_k\).

where

  • \(V\) is a vector space.
  • \(k\) is a positive integer representing the degree of the covector.
  • \(v_1, \dots, v_k \in V\) are vectors.
  • \(A^k(V)\) is the vector space of all \(k\)-covectors on \(V\).
  • \(\text{sgn } \sigma\) is the sign of the permutation \(\sigma\).
  • \(S_k\) is the symmetric group on \(k\) letters.
  • \(\mathbb{R}\) is the set of real numbers.

Note:

  • \(A^k(V)\) is also written \(\Lambda^k(V^*)\).
  • A \(k\)-covector is also called a multicovector of degree \(k\).
  • A \(k\)-covector is also called an alternating \(k\)-tensor.

171.1 Elementary Example

171.1.1 Simple

A \(2\)-covector is an alternating bilinear map on vectors.

\[ \omega : V \times V \rightarrow \mathbb{R} \]

\[ V = \{ e_{1},\ e_{2},\ e_{3} \} \]

\[ \omega(e_{1},e_{2}) = 1,\quad \omega(e_{2},e_{1}) = -1 \]

where

  • \(\omega\) is a \(k\)-covector with \(k = 2\).
  • \(V\) is the set of input basis vectors.