Derivation — Proof of Green's Theorem

The proof of Green's theorem is a masterclass in reducing a two-dimensional identity to the one-dimensional Fundamental Theorem of Calculus, applied twice — once in $x$ and once in $y$. The structure is the template for all the higher-dimensional integral theorems: assume the simplest possible domain (Type I or II), derive the result for that case, and then stitch regions together by cancellation on shared interior boundaries. This note carries out this programme in full, ending with the relationship to the FTC that illuminates why all integral theorems look the same at their core.

• prove Green's theorem for Type I and Type II regions by the FTC argument;
• extend the proof to regions that are both Type I and II;
• understand how Green's theorem for general regions follows by decomposition and cancellation;
• identify the FTC as the zero-dimensional ancestor of Green's theorem.

1. 1. Proof for a Type I Region
2. 1. Proof for a Type II Region
3. 1. Green's Theorem for Simple Regions
4. 1. Extension to General Regions
5. 1. Relationship to the Fundamental Theorem of Calculus