Clifford algebras and multi-dimensional computational differentiation (Primitive introduction to Differential Forms)

The whole physics (or astronomy, engineering, etc.) “lives computationally” in many dimensions. Field theories, mechanics, thermodynamics, etc. operate with gradients, curls, Laplacians, and other geometrical multidimensional differential entities, whose modern incarnations are Forms, such as, e.g., \(dU = A dx dy+ B dy dz\). They (or their coefficients) are endowed with important algebraic/structural properties, and form exterior algebras, not only differential domains. This stuff is useful not only for physicists, and has been applied to robotics, or to the synthesis/analysis of images.

I will show a simple recursive implementation of these entities, using the functional language Haskell; this is a work in progress. I will speak about multi-vectors and their duality properties, and their differentiation. The ambitions of the talk are mainly pedagogical, since the presentation of the Automatic Differentiation techniques is too often restricted to the one-dimensional sector.