Differential Expressions and Their Applications

This paper develops a symbolic framework for differential expressions, treating them as independent entities rather than derivatives of known functions. By introducing nodes and maps, we model the flow and redistribution of differential terms in space, enabling dynamic systems of computation and geometry. Expansion matrices are defined to capture local transformation behavior, and algebraic potential is introduced as a measure of expression change across differential spaces. Applications include modified line integrals, Green’s Theorem under differential control, and image processing on discrete grids using learned coefficient maps.