  We introduce the \emph{$\varphi\mathbb{A}$-differentiability} for functions
$f:U\subset \mathbb R^{k}\to\mathbb A$ where $\mathbb A$ is the linear space
$\mathbb R^{n}$ endowed with an algebra product which is unital, associative,
commutative, $U$ is an open set, and $\varphi:U\subset \mathbb R^{k}\to\mathbb
A$ is a differentiable function in the usual sense. We also introduce the
corresponding generalized Cauchy-Riemann equations ($\varphi\mathbb{A}$-CREs),
the Cauchy-integral theorem, and the \emph{$\varphi\mathbb A$-differential
equations}. The four-dimensional vector fields associated with triangular
billiards are $\varphi\mathbb{A}$-differentiable. The \emph{$\varphi\mathbb
A$-differential equations} can be used for constructing exact solutions of
partial differential equations like the three-dimensional heat equation.
