Green's Theorem: Transforming Integrals in 2D

This lecture introduces Green's Theorem, which allows transforming a 2D integral into a 1D line integral, simplifying computations. The theorem applies to regular domains with positively oriented boundaries. The main idea is to replace a double integral over a surface with a curvilinear integral along the boundary. The lecture covers the interpretation of the theorem, examples of verifying it with different vector fields and domains, and detailed step-by-step computations using polar coordinates to evaluate the integrals. The instructor demonstrates how to apply the theorem to compute the curl of a vector field, integrate over different domains, and verify the equality between the two types of integrals.