Differentiable functions on modules and the equation 𝑔𝑟𝑎𝑑(𝑤)=𝖬𝗀𝗋𝖺𝖽(𝗏)

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.