# Definition Given a $C^\infty$ [[differentiable manifold]] $M$, with a point $P\in M$ and [[tangent space|tangent]] and [[cotangent space|cotangent spaces]] $T_P$ and $T^*_P$, let $\{\pmb{V}_i\}$ and $\{\tilde{\omega}_j\}$ be two sets of $N'$ [[vector field|vector fields]] and $N$ [[one-form field|one-form fields]] respectively. A *differentiable* $C^\infty$ *[[Tensor Field]]* is a function $\bar{F}(\tilde{\omega}_1, \ldots, \tilde{\omega}_N; \pmb{V}_1, \ldots, \pmb{V}_{N'})$ that is $C^\infty$ for any $C^\infty$ [[Differentiable Vector Field|differentiable vector fields]] $\{\pmb{V}_i\}$ and $C^\infty$ [[Differentiable One-Form Field|differentiable one-form fields]] $\{\tilde{\omega}_j\}$.