A Unified System of Tensor Invariants and Their Integral Reduction Theorems

Abstract

This paper organizes the complete operator set and the integral theorems of classical vector analysis into a single unified system. The starting point is algebraic: the second-order tensor invariant of an arbitrary third-order tensor, given by the general epsilon-dual formula (the dual is of order n-p). The paper then investigates the case in which this second-order tensor is the Jacobian matrix of a vector field: there, every classical differential operator is an invariant slice of that one object – the derivative tensor – and every integral theorem is one rung of a single dimensional ladder. On the first floor stands the operator quadruple div–rot–dev–jad (the irreducible decomposition of the derivative tensor, 9=1+3+5 in components), held together by jet, the full second-order tensor invariant; each carries an exact, first-order integral reduction theorem, valid in identical form in R^3, R^4 and R^5. The second floor is built from compositions: the Hessian as jet∘grad, the Laplacian as div∘jet, the Green identities and the Monge-Ampere theorems. Converse theorems (Poincare lemmas, Helmholtz decomposition) sharpen the system into equivalence criteria. As a highlighted application, under the w=ict embedding the jad operator applied to the four-potential yields the electromagnetic field tensor: the homogeneous Maxwell equations and the flux theorem arise as identities, the wave equation and Poynting's theorem as compositions – within this system the sourcelessness of the magnetic field is not a postulate but a theorem. The paper includes the complete R^5 invariant table, jump (boundary-condition) and transport extensions, worked verification examples, the Kaluza-Klein tower of the five-potential, and seven research directions. Section 11.7 looks one floor up: at the fourth-order level the decomposition of the Riemann tensor (20=1+9+10 in d=4: Ricci scalar, traceless Ricci, Weyl) is the counterpart of 9=1+3+5; the two first-order operators of the third-order derivative tensor are the tensor divergence (Cauchy equation) and the tensor curl, whose circuit integral is the Burgers vector; and the continuation of the symmetric sector is the Saint-Venant condition and the Calabi complex. 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.