Introduction into Calculus over Banach algebra

Let $A$, $B$ be Banach $D$-algebras. The map $f:Arightarrow B$ is called differentiable on the set $Usubset A$, if at every point $xin U$ the increment of map $f$ can be represented as $$f(x+dx)-f(x) =frac{d f(x)}{d x}circ dx +o(dx)$$ where $$frac{d f(x)}{d x}:Arightarrow B$$ is linear map and $o:Arightarrow B$ is such continuous map that $$lim_{arightarrow 0}frac{|o(a)|_B}{|a|_A}=0$$ Linear map $displaystylefrac{d f(x)}{d x}$ is called derivative of map $f$. I considered differential forms in Banach Algebra. Differential form $omegainmathcal{LA}(D;Arightarrow B)$ is defined by map $g:Arightarrow Botimes B$, $omega=gcirc dx$. If the map $g$, is derivative of the map $f:Arightarrow B$, then the map $f$ is called indefinite integral of the map $g$ $$f(x)=int g(x)circ dx=intomega$$ Then, for any $A$-numbers $a$, $b$, we define definite integral by the equality $$int_a^bomega=int_{gamma}omega$$ for any path $gamma$ from $a$ to $b$.