Can someone explain the definition of primitive mapping? 
Definition: If $G$ maps an open set $E \subset R^n$ into $R^n$, and if there is an integer $m$ and a real function $g$ with domain $E$ such that $$G(x)=\sum_{i \neq m} x_i e_i +g(x) e_m,\, (x \in E)$$ then we call $G$ primitive.

Can someone explain this definition? How to understand primitive and related equation?

Since not all notions in the question are defined, we try to guess their meaning. We have that $R^n$ probably is $\Bbb R^n$, 
$e_i$ is the standard basis vector of $\Bbb R^n$ such that its $i$-th coordinate is $1$ and the other coordinates are zeroes, given $x\in E\subset \Bbb R^n$, $x=\sum_{i=1}^n x_ie_i$ is the decomposition of $x$ with respect to the basis $\{e_i\}$, that is $x_i$ is the $i$-th coordinate of $x$ for each $1\le i\le n$. Then $G$ is primitive means that $G$ is the identity map on $E$ distorted on $m$-th coordinate for some $1\le m\le n$ by some function $g:E\to\Bbb R$. In particular, if $g(x)=x_m$ for each $x\in E$ then $G$ is the identity map.
