The Unified System of Tensor Invariants — Complete Teaching Edition

Abstract

This edition contains everything the academic article does – the full five-dimensional invariant table, all theorems, the ideas behind the proofs and the research directions – but in teaching language: before every formula it states what the formula will claim and why it is true. The structure follows a building metaphor: the ground floor is the algebra of invariants (the epsilon sorting machine, the third-order tensor up close, the 10=10 duality). The first floor holds the operator quadruple div–rot–dev–jad and jet, with the theorems of the dimensional ladder, jump conditions, the Reynolds transport theorem and worked examples. The second floor is compositions (Hessian = jet∘grad, Laplacian = div∘jet, Green identities, Monge-Ampere). Two side wings carry the determinant theorems (the Piola family, with an explanation of the cofactor matrix) and the converses (Poincare, Helmholtz). The great hall is electrodynamics with boundary conditions, the dynamo and the Kaluza-Klein tower of the five-potential. The roof terrace holds a historical note and seven research directions (among them the fourth-order level: the decomposition of the Riemann tensor, 20=1+9+10 in d=4, and the continuation of the symmetric sector). Boxed intuition asides, six figures and "in plain words" readings support understanding; the closing chapter summarizes the whole system in six sentences. The theorems of the system were also verified symbolically for arbitrary (general polynomial) fields: the published R^4/R^5 invariant tables term by term, the ladder theorems from R^3 to R^5, the Piola family and the null identities – 23 checks, all passed. The material also covers the three-dimensional analogue of the Mobius strip, the solid Klein bottle: there the sector rule becomes component-dependent and the curl splits, its component along the reflected axis staying in sector.