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Mathematics for Physics

165 Differential k-Form

A smooth mapping that assigns an alternating k-linear mapping to each point that is used to integrate over k-dimensional domains.

definition (Differential k-Form) A function assigning an alternating k-linear function to each point of a manifold \(M\), where the following condition applies:

  • For each point \(p \in M\), \(\omega(p)\) is a \(k\)-covector on the tangent space \(T_pM\).

where

  • \(M\) is a smooth manifold.
  • \(k\) is a non-negative integer representing the degree of the form.
  • \(T_pM\) is the tangent space to \(M\) at \(p\).
  • \(T^*_p M\) is the cotangent space at \(p\), defined as the dual space of the tangent space \(T_pM\).
  • \(\Lambda^k(T^*_p M)\) is the vector space of all alternating \(k\)-tensors on \(T_pM\).
  • \(\omega\) is a smooth section of the vector bundle \(\Lambda^k(T^*M)\), the \(k\)-th exterior power of the cotangent bundle.

Note:

  • \(T^*_p M\) is also written \(T^*_p(M)\).
  • \(\Lambda^k(T^*_p M)\) is also written \(A^k(T_pM)\).
  • Alternating \(k\)-tensors are also called \(k\)-covectors.
  • Alternating \(k\)-tensors are also called multicovectors.

165.1 Elementary Example

165.1.1 Simple

A differential \(1\)-form assigns a linear functional to each point of a finite base.

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

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

where

  • \(\omega\) is a differential \(1\)-form.
  • \(M\) is the set of sample points.
  • \(e_{1}\) is a tangent direction at each point.