Section 16.7

Green's Theorem

The 2D Fundamental Theorem of Calculus.

Introduction

Green's Theorem establishes a powerful connection between a Line Integral around a simple closed curve C and a Double Integral over the plane region D bounded by C.

It says: "The spin on the boundary equals the sum of the spins inside."

The Theorem

Statement

Let C be a positively oriented (counter-clockwise), piecewise-smooth, simple closed curve in the plane and let D be the region bounded by C.

This quantity is the "2D Curl" of the field.

Visualizing Macroscopic Circulation

Interactive: Sum of Micro-Spins

Worked Examples

Example 1: Square Path

Evaluate around the square .

1. Without Green's:

Compute 4 line integrals. Slow.

2. With Green's:

3. Solve Double Integral:

Example 2: Area of Ellipse

Find Area of using line integrals.

1. Area Formula:

2. Parameterize Ellipse:

3. Substitute:

.
.

4. Result:

Example 3: Cheat Code

Evaluate on unit circle.

1. Line Integral?

Impossible. Can't integrate .

2. Green's Theorem:

3. Result:

This implies the field is conservative!