Introduction into Calculus over Banach algebra

Let $A$, $B$ be Banach $D$-algebras. The map $f:A\to B$ is called differentiable on the set $U\subset A$, if at every point $x\in U$ the increment of map $f$ can be represented as $$f(x+dx)-f(x)=\frac{df(x)}{dx}\circ dx+o(dx)$$ where $$\frac{df(x)}{dx}:A\to B$$ is linear map and $o:A\to B$ is such continuous map that $$\underset{a\to 0}{\lim }\frac{\mathrm{\parallel }o(a){\mathrm{\parallel }}_B}{\mathrm{\parallel }a{\mathrm{\parallel }}_A}=0$$ Linear map ${\displaystyle \frac{df(x)}{dx}}$ is called derivative of map $f$. I considered differential forms in Banach Algebra. Differential form $\omega \in \mathcal{L}\mathcal{A}(D;A\to B)$ is defined by map $g:A\rightarrow B\otimes B$,$\omega=g\circ dx$\mathrm{.}Ifthemap$g$,isderivativeofthemap$f:A\rightarrow B$,thenthemap$f$iscalledindefiniteintegralofthemap$g$ $$f(x)=\int g(x)\circ dx=\int \omega$$ Then, for any $A$-numbers $a$, $b$, we define definite integral by the equality $${\int }_a^b\omega ={\int }_{\gamma }\omega$$ for any path $\gamma$ from $a$ to $b$.