In Arnold's Ordinary Linear Equations, 1st edition, chapter 2, section 10, subsection 3, the author pulls what is just about the dirtiest, cruelest, most evil trick which could be imagined from an expositor of mathematics. Here is the setup:
"We denote by $F$ the set of all infinitely differentiable functions $f : U \rightarrow \textbf{R}$. Let $\textbf{v}$ be an infinitely differentiable vector field in $U$. The derivative of a function of $F$ in the direction of the field $\textbf{v}$ again belongs to $F$. Thus differentiation in the direction of the field $\textbf{v}$ is a mapping $L_v : F \rightarrow F$ of the algebra of infinitely differentiable functions into itself. Let us consider several properties of this mapping:
- $L_v(f + g) = L_vf + L_vg$
- $L_v(fg) = fL_vg + gL_vf$
- $L_{u+v} = L_v + L_u$
- $L_{fu} = fL_u$
- $L_uL_v = L_vL_u$
($f$ and $g$ are smooth functions and $\textbf{u}$ and $\textbf{v}$ are smooth vector fields)."
And then comes the punch to the gut, in tiny text:
"Problem 1. Prove properties 1-5, except for the one which is not true."
I believe the falsehood is number 4, $L_{fu} = fL_u$. The reason is because $f$ must be a function from a vector field to another vector field to be applied to $u$, but $L_u$ transforms a real-valued function to another real-valued function, so it doesn't make sense to apply the same $f$ to both $u$ and $L_u$.
Is that correct?