  Let $F(z)=z-H(z)$ with order $o(H(z))\geq 1$ be a formal map from $\bC^n$ to
$\bC^n$ and $G(z)$ the formal inverse map of $F(z)$. We first study the
deformation $F_t(z)=z-tH(z)$ of $F(z)$ and its formal inverse
$G_t(z)=z+tN_t(z)$. (Note that $G_{t=1}(z)=G(z)$ when $o(H(z))\geq 2$.) We show
that $N_t(z)$ is the unique power series solution of a Cauchy problem of a PDE,
from which we derive a recurrent formula for $G_t(z)$. Secondly, motivated by
the gradient reduction obtained by M. de Bondt, A. van den Essen \cite{BE1} and
G. Meng \cite{M} for the Jacobian conjecture, we consider the formal maps
$F(z)=z-H(z)$ satisfying the gradient condition, i.e. $H(z)=\nabla P(z)$ for
some $P(z)\in \bC[[z]]$ of order $o(P(z))\geq 2$. We show that, under the
gradient condition, $N_t(z)=\nabla Q_t(z)$ for some $Q_t(z)\in \bC[[z, t]]$ and
the PDE satisfied by $N_t(z)$ becomes the $n$-dimensional inviscid Burgers'
equation, from which a recurrent formula for $Q_t(z)$ also follows.
Furthermore, we clarify some close relationships among the inversion problem,
Legendre transform and the inviscid Burgers' equations. In particular the
Jacobian conjecture is reduced to a problem on the inviscid Burgers' equations.
Finally, under the gradient condition, we derive a binary rooted tree expansion
inversion formula for $Q_t(z)$. The recurrent inversion formula and the binary
rooted tree expansion inversion formula derived in this paper can also be used
as computational algorithms for solutions of certain Cauchy problems of the
inviscid Burgers' equations and Legendre transforms of the power series $f(z)$
of $o(f(z))\geq 2$.
