Let $\sigma \in \mathfrak{X}(\Omega)$ be a fixed smooth vector field on an open domain $\Omega \subset \mathbb{R}^d$. Consider the operator $T\colon C^\infty(\Omega) \to C^\infty(\Omega)$ on the space of smooth scalar fields $f\colon \Omega \to \mathbb R$ given by $Tf := \nabla f \cdot \sigma$. Is there a simple expression for the Fréchet derivative of $T$? (or perhaps some other sensible interpretation of the derivative of $T$).
Since $T$ is linear the basepoint of the derivative should not matter. I also have a suspicion that it might be given by pointwise mutliplication with $-\operatorname{div} \sigma$, but I am not sure. I am only concerned with the cases $\Omega \in \{\mathbb R^d, (0,1)^d\}$ for $d=2,3$, and I also do not care about regularity conditions, but I do not think it will be relevant.
As a remark note that $Tf = \langle \nabla f , \sigma\rangle = df(\sigma) = \sigma(f)$ by unpacking definitions.
Concerning the Frechet derivative of $T$, note that $T$ is a continuous linear map and as such equal to its own derivative (remember that the derivative is the linear part of the best affine approximation of a map, hence the derivative of any continuous linear map must be this linear map itself).