TY - BOOK

T1 - Wedge product of submanifold distributions with applications to classical electromagnetism

AU - Banaszek, Thomas

PY - 2018

Y1 - 2018

N2 - In this thesis we propose a new and novel approach to expressing distributions on submanifolds. To this end we introduce a set of objects called submanifold distributions constructed out of the embedding maps and the tools of exterior calculus. We then define a multiplication between these objects called the wedge product. Both of the aforementioned are succinctly defined without reference to a coordinate system. Following with the coordinate free theme we also define the pullback of a submanifold distribution and show that this provides a powerful framework for calculating solutions to linear differential equations via Green's methods. We then investigate multipoles in terms of submanifold distributions and find an unusual result for quadrupoles. It is remarkable that quadrupoles have been extensively studied for over a century yet there is no mention of quadrupole transformations between general coordinate systems. More so, we show that quadrupoles do not in fact transform as a tensor, as suggested by the literature, but instead possess more complicated transformation rules that involve second order derivatives and an integral. We conclude this thesis with the calculation of the Lienard-Wiechert field for moving dipoles and quadrupoles using the newly developed machinery of wedge products and submanifold distributions.

AB - In this thesis we propose a new and novel approach to expressing distributions on submanifolds. To this end we introduce a set of objects called submanifold distributions constructed out of the embedding maps and the tools of exterior calculus. We then define a multiplication between these objects called the wedge product. Both of the aforementioned are succinctly defined without reference to a coordinate system. Following with the coordinate free theme we also define the pullback of a submanifold distribution and show that this provides a powerful framework for calculating solutions to linear differential equations via Green's methods. We then investigate multipoles in terms of submanifold distributions and find an unusual result for quadrupoles. It is remarkable that quadrupoles have been extensively studied for over a century yet there is no mention of quadrupole transformations between general coordinate systems. More so, we show that quadrupoles do not in fact transform as a tensor, as suggested by the literature, but instead possess more complicated transformation rules that involve second order derivatives and an integral. We conclude this thesis with the calculation of the Lienard-Wiechert field for moving dipoles and quadrupoles using the newly developed machinery of wedge products and submanifold distributions.

U2 - 10.17635/lancaster/thesis/483

DO - 10.17635/lancaster/thesis/483

M3 - Doctoral Thesis

PB - Lancaster University

ER -