Let \(M\) be a smooth manifold and \(F\in \Gamma(T^{(k,l)}TM)\) a smooth tensor field . Then the connection defines a smooth \((k,l+1)\)-tensor field
\begin{equation*} \nabla F\colon \underbrace{\mathcal{T}^1(M)\times \cdots \times \mathcal{T}^1(M)}_\text{\(k\) factors}\times \underbrace{\mathfrak{X}(M)\times \cdots \mathfrak{X}(M)}_\text{\(l+1\) factors} \to C^\infty(M), \end{equation*}where the last argument defines the direction of the covariant derivative. This tensor is called total covariant derivative of \(F\).