Encoding Electromagnetic Transformation Laws for Dimensional Reduction

Abstract

Electromagnetic phenomena are mathematically described by solutions of boundary value problems. For exploiting symmetries of these boundary value problems in a way that is offered by techniques of dimensional reduction, it needs to be justified that the derivative in symmetry direction is constant or even vanishing. A generalized notion of symmetry can be defined with different directions at every point in space, as long as it is possible to exhibit unidirectional symmetry in some coordinate representation. This can be achieved, e.g., when the symmetry direction is given by the direct construction out of a unidirectional symmetry via a coordinate transformation which poses a demand on the boundary value problem. Coordinate independent formulations of boundary value problems do exist but turning that theory into practice demands a pedantic process of backtranslation to the computational notions. This becomes even more challenging when multiple chained transformations are necessary for propagating a symmetry. We try to fill this gap and present the more general, isolated problems of that translation. Within this contribution, the partial derivative and the corresponding chain rule for multivariate calculus are investigated with respect to their encodability in computational terms. We target the layer above univariate calculus, but below tensor calculus.