On page 325 of Stein's Functional Analysis, he writes "We consider the following vector field
$$ L = \frac{1}{i\lambda} \sum_{k=1}^d a_k \frac{\partial}{\partial x_k} = \frac{1}{i\lambda}(a \cdot \nabla) $$
... then the transpose $L^t$ of $L$ is given by
$$ L^t(f) = -\frac{1}{i\lambda} \sum_{k=1}^d \frac{\partial}{\partial x_k} (a_k f) = -\frac{1}{i\lambda} \nabla \cdot (af) $$
..." (Here, $f$ is a function $\mathbb R^d \to \mathbb R$, and $a$ is a function $\mathbb R^d \to \mathbb R^d$.)
Can someone explain to me where the expression for the transpose comes from?
If I understand correctly, $L^t$ and $L$ are related by
$$ \int_{\mathbb R^d} L(f) g = \int_{\mathbb R^d} f L^t(g) $$
In an attempt to show that the two sides are equal, I made the calculation
$$ \int_{\mathbb R^d} L(f) g - \int_{\mathbb R^d} f L^t(g) = \int_{\mathbb R^d} \nabla \cdot (afg) $$
However, I am not sure what to do from here. Can someone help?
