Generalizing Multivariable Convergence: From Polar Transformations to Complex-Dual Structures

This 55-page research provides a comprehensive development of multivariable convergence theories. Starting from foundational principles and classical polar transformations, the work systematically constructs a new mathematical framework using Complex-Dual numbers. By analyzing limits and continuity through this expanded number system, I derive several new theorems and rigorous proofs that generalize traditional calculus. The research bridges the gap between basic multivariable analysis and advanced dual-plane structures, offering fresh insights into function behavior at the origin and beyond.