A Unified System of Tensor Invariants and Their Integral Reduction Theorems

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 epsilon -dual of a tensor of order p in n dimensions, an order-lowering map to order n-p . The analytic part rests on a second and structurally opposite object, the order-raising total derivative: for a vector field this is the derivative tensor, every classical differential operator is an invariant slice of it, and every integral theorem is one rung of a single dimensional ladder. Keeping the two apart — one axis built from epsilon , the other from the metric — is what lets the symmetric and the antisymmetric sector be treated together. On the first floor stands the operator quadruple div–rot–dev–jad. The decomposition behind it has three slices — trace, antisymmetric and symmetric-traceless, 9=1+3+5 in components — with rot and jad the two readings of the antisymmetric one. Each slice carries an exact, first-order integral reduction theorem — for the antisymmetric slice in its jad form, of which rot is the three-dimensional dual reading — valid in identical form in R^3 , R^4 and R^5 . A completeness theorem shows that within the scope of the master identity this list is exhaustive: for a vector field there are exactly three rotationally natural such theorems and no fourth — except in four dimensions, where the orientation group sees the jad theorem split into self-dual halves. The second floor is built from compositions: the Hessian as grad∘grad, the Laplacian as div∘grad, 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, once the potential is taken as the primitive variable, the sourcelessness of the magnetic field follows as an identity rather than being posited, and the physical content is carried by the converse: a magnetic monopole would mean precisely that the field tensor cannot be written as jad of a potential. The claims of the system are checked by the scripts shipped with the article: symbolically, over whole classes of polynomial fields, and numerically, where the runs screen for convention and transcription errors rather than prove; the Appendix separates the two. Numerical values quoted from the companion works [19] and [21] are attributed there.