Teorema de Green: uma prova geral e as generalizações do cálculo vetorial

Abstract

This work aims to provide a rigorous proof of Green’s Theorem for regions determined by rectifiable Jordan curves in the plane. To this end, we first establish the foundations of Vector Analysis, formalizing the concepts of vector fields, rectifiable curves, curve orientation, and Jordan regions. We then develop the necessary integration theory, covering the classical Riemann integral, multiple integrals, line integrals for scalar and vector fields, and their fundamental properties of additivity, reparametrization invariance, and iterated integration. With this foundation, we prove Green’s Theorem first for rectangular regions and then for the general case of rectifiable Jordan regions, employing a grid subdivision technique that allows the region to be decomposed into controlled subregions. Finally, we use Green’s Theorem as a tool to prove particular cases of Stokes’ Theorem and look for a practical application in a problem of calculus of variations.