Functional Coding of Differential Forms

Algebraic computations in differential geometry have usually a strong “analytic” side, and symbolic formula crunching is heavily used, even if at the end, the user needs only numbers, or graphic visualization. We show how to implement in a simple way the domain of differential forms with the p-vector algebra, Hodge “star” operator, and the differentiation. There is no explicit symbolic manipulation involved, we exploit only the “standard” mathematical operations in a generic way. Everything forms a local algebra coded in Haskell, and the differentiation algorithms heavily use the lazy evaluation. Some short examples are presented. This paper generalizes our one-dimensional algorithmic differentiation formalism in functional sauce presented elsewhere.