I was learning differential topology, here is one of the definitions of Jet bundle on the book :Stable Mappings and Their Singularities by M.Golubitsky & V. Guillemin,as follows:
Definition 2.1. Let $X$ and $Y$ be smooth manifolds, and $p$ in $X$. Suppose $f, g: X \rightarrow Y$ are smooth maps with $f(p)=g(p)=q$.
(1) $f$ has first order contact with $g$ at $p$ if $(d f)_{p}=(d g)_{p}$ as mapping of $T_{p} X \rightarrow T_{q} Y$
(2) $f$ has $k$ th order contact with $g$ at $p$ if $(d f): T X \rightarrow T Y$ has $(k-1) s t$ order contact with $(dg)$ at every point in $T_{p} X .$ This is written as $f \sim_{k} g$ at $p .$ ( $k$ is a positive integer.)
What I don't understand is the $k\ge 2$ order case(say for example $k =2$),since by definition it means $df$ has first order contact with $dg$ at $p$,which means by (1) $(ddf)_p = (ddg)_p$ but we know $d^2 = 0$ why can we do like this?