One-forms¶

Consider \(T_P\), the tangent space of vectors at a point \(P\). A one-form, \(\of{\omega}\) at \(P\) associates a vector \(\vec{V}\) at P to a real number, \(\of{\omega}(\vec{V})\).

This function is linear; for vectors \(\vec{V}\) and \(\vec{W}\), and real numbers \(a\) and \(b\),

\[\label{eq:8} \of{\omega}( a \vec{V} + b \vec{W} ) = a \of{\omega}(\vec{V}) + b \of{\omega}(\vec{W})\]

can be multiplied by scalars,

\[\label{eq:9} (a \of{\omega})(\vec{V}) = a [ \of{\omega}(\vec{V})]\]

and have the property

\[\label{eq:10} (\of{\omega} + \of{\sigma}) (\vec{V}) + \of{\omega}(\vec{V}) + \of{\sigma}(\vec{V})\]

They satisfy the axioms of a vector space, and are indeed the duals of vectors, and have a tangent vector space \(T^{*}_P\).

A field of one-forms can represent the gradient of a function \(f\); such a field is denoted \(\dd{f}\),

\[\of{\dd{}}f (\dv*{\lambda}) = \dv{f}{\lambda}\]

In the vector space \(T^{*}_P\) any \(n\) linearly independent one-forms can constitute a basis, however selecting a set of basis vectors, \(\set{\vec{e}_i}\) in \(T_P\) induces a preferred basis on \(T^{*}_P\), the dual basis \(\set{\of{\omega}^i}\).

These have the property

\[\of{\omega}^i(\vec{e}_j) = \delta^i_j\]