# Definition Given a [[Differentiable Manifold|differentiable]] $C^\infty$ [[manifold]] $M$, with a point $P\in M$ and [[Tangent Space|tangent]]and [[Cotangent Space|cotangent spaces]] $T_P$ and $T^*_P$, let $\pmb{V}$ and $\tilde{\omega}$ be a [[vector field]] and a [[One-Form Field]] respectively. A *differentiable* $C^\infty$ *[[One-Form Field]]* is a function $\tilde{\omega}(\pmb{V})$ that is $C^\infty$ for any $C^\infty$ [[differentiable vector field]] $\pmb{V}$ One can see that this notion can be defined in terms of [[components]] as well. That is, $\tilde{q}$ is a differentiable $C^\infty$ [[one-form field]] if and only if its [[components]] $\{q^i\}$ associated with a $C^\infty$ [[basis]] for [[Vector Field|vector fields]] are $C^\infty$ [[Differentiable Function|differentiable funtions]].