series $ D_{i}F\in B[t_{1},\dots ,t_{n})\otimes B[t_{1},\dots ,t_{n}] $. Here $ D_{i} $ is defined by the following properties; here we think of $ B[t_{1},\dots ,t_{n}] $ and $ B[t_{1},\dots ,t_{n})\otimes B[t_{1},\dots ,t_{n}] $ are viewed as bimodules over the algebra generated by $ B $ and $ t_{1},\dots ,t_{n} $ using its obvious left and right actions.

1.  $ D_{i} $ is bilinear over the algebra generated by $ B $ and $ t_{1},\dots ,t_{i-1},t_{i+1},\dots ,t_{n} $;

2.  $ D_{i}(t_{i})=1\otimes 1 $;

3.  $ D $ satisfies the Leibniz rule: $ D_{i}(FG)=(D_{i}F)G+F(D_{i}G) $.

As an example, $$D_{1}(b_{0}t_{1}b_{1}t_{2}b_{2}t_{1}b_{3}t_{4}b_{4})=b_{0}\otimes b_{1}t_{2}b_{2}t_{1}b_{3}t_{4}b_{4}+b_{0}t_{1}b_{1}t_{2}b_{2}\otimes b_{3}t_{4}b_{4}.$$

Given a family of non-commutative power series $ F_{1},\dots ,F_{n} $ with a common multiradius of convergence $ (R_{1},\dots ,R_{n}) $, we define for $ {X_{1},\dots ,X_{n}}\in M $, $ \Vert X_{i}\Vert <R_{i} $, its Jacobian at $ {X_{1},\dots ,X_{n}}$, to be the matrix $ D_{B}F(X_{1},\dots ,X_{n})\in M_{n}\otimes M\otimes M $ whose $ i,j $-th entry is equal to $ D_{i}(F_{j})({X_{1},\dots ,X_{n}}) $.

Note that if $ B\subset M_{k} $ and $ {x_{1},\dots ,x_{n}}\in M_{k} $, then $ D_{B}({x_{1},\dots ,x_{n}}) $ is precisely the Jacobian of the map $$(z_{1},\dots ,z_{n})\mapsto \left( F_{1}(z_{1},\dots ,z_{n}),\dots ,F_{n}(z_{1},\dots ,z_{n})\right) ,$$ evaluated at $ {x_{1},\dots ,x_{n}}$.

