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 a 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 decomposition of the derivative tensor into trace, antisymmetric and symmetric-traceless parts, 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–Ampère theorems. Converse theorems (Poincaré lemmas, Helmholtz decomposition) sharpen the system into equivalence criteria, each rung with its own topological condition. 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, a consequence of taking the potential as the primitive variable. The claims of the system are verified by the symbolic and numerical scripts shipped with the article; the numerical values quoted from the companion works [19] and [21] are attributed there.