Authors: J.A.J. van Leunen
Two quite different forms of differential calculus exist that both have physical significance. The most simple version is quaternionic differential calculus. Maxwell based differential calculus is based on the equations that Maxwell and others have developed in order to describe electromagnetic phenomena. Both approaches can be represented by four-component “fields” and four-component differential operators. Both approaches result in a dedicated non-homogeneous second order partial differential equation. These equations differ and offer solutions that differ in details. Maxwell based differential calculus uses coordinate time t, where quaternionic differential calculus uses proper time τ. The consequence is that also the interpretation of speed differs between the two approaches. A more intriguing fact is that these differences involve a different space-progression model and different charges and currents. The impacts of these differences are not treated in this paper. By adding an extra Maxwell based differential equation the conformance between the two approaches increases significantly. The formulation of physics in Maxwell based differential calculus differs significantly from the formulation of physics in quaternionic differential calculus. It results in a different space-progression model. The choice between the two approaches influences the description of physical reality. However, the selected formulation does not affect physical reality. The description does not affect the described field. The conclusion of the paper is that depending on the type of investigated phenomena either the Maxwell based approach or the quaternionic approach fits better as a descriptor. The Maxwell based approach fits better for describing wave behavior. The quaternionic approach fits better for the description of the embedding process. Quaternionic differential calculus also fits better with the application of Hilbert spaces in quantum physics than Maxwell based differential calculus does. However, Maxwell based differential calculus is the general trend in current physical theories.
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