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

Published online by Cambridge University Press:  14 November 2011

William C. Waterhouse
William C. Waterhouse
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, U.S.A.
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Synopsis

Let 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.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 122 , Issue 3-4 , 1992 , pp. 353 - 361
Copyright
Copyright © Royal Society of Edinburgh 1992

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