Abstract
The purpose of this paper is to prove that most of the differential geometry of mathematical physics today is based on a confusion between two differential sequences, namely, the Janet sequence (1920) and the Spencer sequence (1970) that can be found in the formal theory of systems of partial differential equations and Lie pseudogroups. Correcting this confusion by means of new homological methods (diagram chasing) brings at once the need for visiting again the foundations of well established theories, including continuum mechanics, gauge theory and general relativity. In particular, we give for the first time a modern exposition of E. and F. Cosserat elasticity theory along the work of D. C. Spencer, while extending it to electromagnetism along the idea of H. Weyl for using the conformal group of space-time. Finally, exactly as Newton theory of gravitation is not coherent with pseudogroup theory. More precisely, Yang-Mills equations and Einstein equations are not coherent with Spencer sequence!
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