On the Concept of Oriented Operators on Smooth Manifolds

This research explores a pivotal concept in the structure of space, facilitating the definition and operation of oriented operators. These operators serve as a crucial tool for describing algebraic varieties in higher-dimensional spaces. The paper delves into the concept's intrinsic relationship with advanced mathematical frameworks, including differential geometry, functional analysis, and topology. It offers a systematic exposition, starting with foundational examples in calculus and differential geometry in Section 2, and culminating in the formal introduction and analysis of the concept in Section 3. The study suggests that this concept, though fundamental, has been implicitly utilized in existing mathematical discourse without explicit acknowledgment.