Let
A
A
be a finite-dimensional, commutative algebra over
R
\mathbb {R}
or
C
\mathbb {C}
. The notion of
A
A
-differentiable functions on
A
A
is extended to develop a theory of
A
A
-differentiable functions on finitely generated
A
A
-modules. Let
U
U
be an open, bounded and convex subset of such a module. An explicit formula is given for
A
A
-differentiable functions on
U
U
of prescribed class of differentiability in terms of real or complex differentiable functions, in the case when
A
A
is singly generated and the module is arbitrary and in the case when
A
A
is arbitrary and the module is free. Certain components of
A
A
-differentiable function are proved to have higher differentiability than the function itself.
Let
M
\mathsf {M}
be a constant, square matrix. By using the formula mentioned above, a complete description of solutions of the equation
grad
(
w
)
=
M
grad
(
v
)
\operatorname {grad}(w)=\mathsf {M}\operatorname {grad}(v)
is given.
A boundary value problem for generalized Laplace equations
M
∇
2
v
=
∇
2
v
M
⊺
\mathsf {M}\nabla ^2 v=\nabla ^2v \mathsf {M}^{\intercal }
is formulated and it is shown that for given boundary data there exists a unique solution, for which a formula is provided.