The Unified System of Tensor Invariants — Complete Teaching Edition

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 six research directions. 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.