Project Title:
Axioms of Calculus and Mechanics
Summary:
We present axioms of calculus that unify the discrete and the smooth continuum with applications to fractals, soap films, charged particles, and smooth manifolds. Properties that follow from the axioms include broad generalizations of the integral theorems of calculus that are equally valid for the above domains. A goal for the period of this grant is to establish similar axioms of mechanics that unify the quantum and classical viewpoints.
Technical Abstract:
We present axioms of calculus that unify the discrete and the smooth continuum with applications to fractals, soap films, charged particles, and smooth manifolds. Properties that follow from the axioms include broad generalizations of the integral theorems of calculus that are equally valid for the above domains. Emerging from the axioms is a new derivative and integral called prederivative and preintegral. The first takes on the role of weak derivative of distributions and permits us to multiply singular distributions without a contradiction of mathematics. Preintegral together with prederivative suggest a way to formalize conservation of energy in a curved space. Hodge decomposition in our category leads to rigorous proof of the existence of the Coulomb field and Maxwell's equations. A similar proof shows how 2-dimensional shell fields form a dynamic fabric of space. We find new representations of the CCR with physical realism using prederivative and its adjoint. A goal for the period of this grant is to establish similar axioms of mechanics that unify the quantum and classical viewpoints.
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