A multivector data structure for differential forms and equations
We use tools from algebraic topology to show that a class of structural differential equations may be represented combinatorially, and thus, by a computer data structure. In particular, every differential k-form may be represented by a formal k-cochain over a cellular structure that we call a starplex, and exterior differentiation is equivalent to the coboundary operation on the corresponding k-cochain. Furthermore, there is a one-to-one correspondence between this model and the classical finite cellular model supported by the Generalized Stokes' Theorem and translation between the two models can be completely automated. Our results point the way to a common combinatorial and data structure well-suited for a physical modeling computer algebra that unifies finite and infinitesimal, symbolic and numeric, geometric and physical descriptions of distributed phenomena. We illustrate the advantages of our approach by a prototype interactive physics editor that uses the computer algebra to automatically translate intuitive geometrical/physical descriptions of balance conditions created by the user into the corresponding symbolic differential and integral equations. (C) 2000 IMACS. Published by Elsevier Science B.V. All rights reserved.
multivector data structure differential equations k-cochain combinatorial topology