We use tools from algebraic topology to show that a class of differential equations may be represented combinatorially, and 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, with names of the cells chosen to correspond to a set of coordinate basis forms. The resulting data structure combines differential forms of all dimensions and matches exactly the combinatorial structure of multivectors in geometric algebra; it also provides a natural geometric interpretation to many common symbolic operations. For example, exterior differentiation is equivalent to a modified coboundary operation on the corresponding k-cochain.

We illustrate the advantages of our approach by a prototype interactive physics editor that automatically translates intuitive geometrical/physical descriptions of physical laws created by the user into the corresponding symbolic differential and equations, and vice versa. The correctness of the translations is assured by the Generalized Stokes’ Theorem and its dual.

vshapiro@engr.wisc.edu