Let $U\subseteq\mathbb{R}^n, V\subseteq\mathbb{R}^m$ be open subsets, and $f:U\rightarrow V$ a $C^1$ map. Let $u,v$ be vector fields on $U,V$ respectively. Show that $u,v$ are $f$-related if and only if $$L_uf^*\varphi=f^*L_v\varphi$$ for every $\varphi\in C^1(V)$.
I am confused about the notation $f^*$. In my notes there is the symbol $f_*v$, which denotes the push-forward of $v$ by $f$. By definition, it is the vector field $w$ on $V$ such that $v$ and $w$ are $f$-related.
What does $f^*$ mean here? Is it just a typo that should be $f_*$, or is it something else?
$f^*$ is the pull back. For scalar fields, it is defined as $f^*\varphi = (x \mapsto\varphi(f(x)))$. (It can also be defined for one forms as the formal adjoint of $f_*$.)