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Mathematics for Physics

172 k-Covector Field

A smooth mapping that assigns a k-covector to each point that is used to define differential forms.

definition (k-Covector Field) A function assigning a \(k\)-covector to each point of a manifold, \(\omega: M \rightarrow \Lambda^k(T^*M)\), where the following condition applies:

  • For each point \(p\) in \(M\), \(\omega(p)\) is an alternating \(k\)-linear function on the tangent space \(T_pM\).

where

  • \(M\) is a smooth manifold.
  • \(p\) is a point in \(M\).
  • \(\omega\) is the \(k\)-covector field.
  • \(T_pM\) is the tangent space of \(M\) at \(p\).
  • \(T^*_pM\) is the cotangent space of \(M\) at \(p\).
  • \(\Lambda^k(T^*_pM)\) is the space of all alternating \(k\)-tensors on \(T_pM\).
  • \(\Lambda^k(T^*M)\) is the \(k\)-th exterior power of the cotangent bundle.

Note:

  • \(\Lambda^k(T^*_pM)\) is also written \(A^k(T_pM)\).
  • A \(k\)-covector field is also called a differential \(k\)-form.

172.1 Elementary Example

172.1.1 Simple

A \(k\)-covector field assigns a \(k\)-covector to each point. Here \(k = 1\) on three points.

\[ M = \{ p,\ q,\ r \} \]

\[ \omega(p)(e_{1}) = 1,\quad \omega(q)(e_{1}) = 0,\quad \omega(r)(e_{1}) = 2 \]

where

  • \(\omega\) is the \(k\)-covector field.
  • \(M\) is the set of points.
  • \(\omega(p)\) is the covector at \(p\).