Let \(M\) be a smooth manifold. Then a linear map
\begin{equation*} F\colon \underbrace{\mathcal{T}^1\times \cdots \times \mathcal{T}^1}_\text{\(k\) factors} \times \underbrace{\mathfrak{X}(M)\times \cdots \times \mathfrak{X}(M)}_\text{\(l\) factors} \to C^\infty(M), \end{equation*}is a smooth tensor field if and only if it is \(C^\infty (M)\)-multilinear.
In particular, the value of an \(C^\infty (M)\)-multilinear map in a particular point \(p\in M\) depends only on the values of its arguments in \(p\).