  Let $A$ be a finite-dimensional, commutative algebra over $\mathbb{R}$ or
$\mathbb{C}$. The notion of $A$-differentiable functions on $A$ is extended to
the notion of $A$-differentiable functions on a finitely generated $A$-module
$B$. Let $U$ be an open, bounded and convex subset of $B$. When $A$ is singly
generated and $B$ is arbitrary or $A$ is arbitrary and $B$ is a free module, an
explicit formula for an $A$-differentiable functions on $U$, of a prescribed
class of differentiability, is given in terms of real or complex differentiable
functions. It appears, even in case of real algebras, that certain components
of $A$-differentiable function are of higher differentiability than the
function itself.
  Let $M$ be a constant, square matrix. Using the aforementioned formula we
find the complete solution of the equation $grad(w)=Mgrad(v)$.
  The boundary value problem for generalized Laplace equations $M\nabla^2
v=\nabla^2v M^{\intercal}$ is formulated and it is proved that for the given
boundary data there exists an unique solution, for which a formula is provided.
