Midpoint Partial Calculus: Multidimensional Calculus and Electromagnetism 中道偏微分 on Non-Associative Algebras

Abstract This paper extends Midpoint Calculus from one dimension to multiple dimensions, establishing the Midpoint Partial Calculus (MPC) theoretical framework. On the commutative non-associative algebra C³ and the non-commutative non-associative algebra B6, we systematically verify six directions: 2D integration, Green's theorem, wave equation, radial recovery theorem, 3D Stokes theorem, and midpoint electromagnetism. Core findings include: (1) the associator-deviation identity holds exactly in 1D/2D/3D; (2) domain additivity corresponds to deviation cohomology class [δ] ∈ Hⁿ(𝒜, Region); (3) non-commutativity introduces "commutation ambiguity", splitting electromagnetic constitutive relations into two versions; (4) the Poynting vector naturally arises as the commutator [E, H], where non-commutativity itself becomes energy flux. Keywords: non-associative algebra, midpoint calculus, partial differential equations, Green's theorem, Stokes theorem, electromagnetism, B6 algebra