  The Spencer operator, introduced by D.C. Spencer fifty years ago, is rarely
used in mathematics today and, up to our knowledge, has never been used in
engineering applications or mathematical physics. The main purpose of this
paper, an extended version of a lecture at the second workshop on Differential
Equations by Algebraic Methods (DEAM2, february 9-11, 2011, Linz, Austria) is
to prove that the use of the Spencer operator constitutes the common secret of
the three following famous books published about at the same time in the
beginning of the last century, though they do not seem to have anything in
common at first sight as they are successively dealing with elasticity theory,
commutative algebra, electromagnetism and general relativity: (C) E. and F.
COSSERAT: "Th\'eorie des Corps D\'eformables", Hermann, Paris, 1909. (M) F.S.
MACAULAY: "The Algebraic Theory of Modular Systems", Cambridge University
Press, 1916. (W) H. WEYL: "Space, Time, Matter", Springer, Berlin, 1918 (1922,
1958; Dover, 1952). Meanwhile, we shall point out the importance of (M) for
studying control identifiability and of (C)+(W) for the group theoretical
unification of finite elements in engineering sciences, recovering in a purely
mathematical way well known field-matter coupling phenomena (piezzoelectricity,
photoelasticity, streaming birefringence, viscosity, ...). As a byproduct and
though disturbing it could be, we shall prove that these unavoidable new
diferential and homological methods contradict the mathematical foundations of
both engineering (continuum mechanics,electromagnetism) and mathematical (gauge
theory, general relativity) physics.
