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The purpose of this short notice is to present an elementary summary of a fewrecent results obtained through the application of the formal theory of systemsof partial differential equations and Lie pseudo groups to engineering(elasticity theory, electromagnetism, coupling phenomena) and mathematical(gauge theory, general relativity) along the following scheme: 1) Lie groups oftransformations may be considered as Lie pseudo groups of transformations butno action type method can be used as parameters never appear any longer. 2) Thework of Cartan is superseded by the use of the canonical Spencer sequence whilethe work of Vessiot is superseded by the use of the canonical Janet sequencebut the link between these two sequences and thus these two works is not knowntoday. 3)Using duality theory, the formal adjoint of the Spencer operator forthe conformal group of transformations of space-time provides the Cosseratequations, the Maxwell equations and the Weyl equations on equal footing butsuch a result leads to deep contradictions. Accordingly, the results thusobtained prove that the foundations of engineering and mathematical physicsmust be revisited within the framework of jet theory though striking it maylook like for established theories.

A pedestrian approach to cosserat / Maxwell / weyl theory