Differentiable functions on algebras and the equation grad (w) = M grad (v)

SynopsisLet U be a convex open set in a finite-dimensional commutative real algebra A. Consider A-differentiable functions f: U → A. When they are C2 as functions of their real variables, their A-derivatives are again A-differentiable, and they have second-order Taylor expansions. The real components of such functions then have second derivatives for which the A-multiplications are self-adjoint. When A is a Frobenius algebra, that condition (a system of second-order differential equations) actually forces a real function on U to be a component of some such f. If v is a function of n real variables, and M is a constant matrix, then the requirement that M∇(u) should equal ∇(w) for some w usually falls into this setting for a suitable A, and the quite special properties of such v, w can be deduced from known properties of A-differentiable functions.