Abstract

In this three part investigation we provide the mathematical foundations and principles of a frame indifferent classical electromagnetic field theory (FIEFT) for arbitrarily moving material media with arbitrary constitution based on convective and comoving time derivative operators. Part 1 is devoted to the mathematical tools utilized in establishing the field theory. It starts with the description of material points in arbitrary Euclidean motion, which is a characteristic of rigid (non-deforming) bodies and incompressible inhomogeneous fluids in continuum mechanics. Next we establish the mathematical link between spatial and time derivatives of vector fields between Eulerian and Lagrangian frames via coordinate transformations in Euclidean space. Regarding the images of time derivatives of field quantities, we necessarily invoke the convective and comoving time derivatives. We also provide a proof of the representation of the comoving time derivative for scalar and vector density fields along with its certain differential, commutative and integral properties. In Part 2 we provide the axiomatic structure of our field theory where the frame indifferent electromagnetic field equations are obtained directly as images of Maxwell equations of stationary media under Euclidean (aka observer) transformations. The commutative properties derived between spatial differential and comoving time derivative operators help us derive progressive wave equations for the two standard (translational and rotational) types of Euclidean motion. In Part 3 we describe the general formulation of a boundary value problem for an arbitrarily moving object and investigate three canonical problems of practical interest to demonstrate the predictions of FIEFT.