Integration over manifolds

Question

Let $M$ be a manifold and $I$ a continuous linear application of the smooth functions over $M$, ${\cal C}^{\infty}(M)$. I suppose that:

$$I(f).I(g)=I(I(f).g)+I(f.I(g))$$

with $f,g \in {\cal C}^{\infty}(M)$.

Then, have we:

$$I(f)=\int_{-\infty}^0 e^{tX}(f)dt$$

where $X$ is a vector fiels over $M$?