Green’s theorem on rectangles

This video provides a visual proof of Green's Theorem using the "small rectangle method." The proof follows four logical steps: partitioning, approximation, summation, and taking limits. First, a planar region D is divided into a fine grid of infinitesimal rectangles. Next, the line integral around a single small rectangle is approximated using first-order Taylor expansions of partial derivatives, yielding an area integral form. Then, summing the integrals over all sub-rectangles reveals that internal shared boundaries cancel out due to opposite traversal directions, leaving only a staircase-like outer boundary. Finally, as the grid size approaches zero, this staircase boundary converges to the original smooth curve L, and the discrete Riemann sum transforms into a double integral, rigorously deriving Green's Theorem.

Before you watch

• Fundamental Theorems of Multivariable Calculus
• Concept of Partial Derivatives and Taylor Series Approximation
• Definitions and Orientation Properties of Line Integrals
• Concepts of Double Integrals and Riemann Sums