Green's Theorem
Green's theorem relates a line integral around a closed curve to a double integral over the enclosed region.
A fundamental result in vector calculus.
The Formula
Green's theorem converts a line integral around a closed curve C into a double integral over the region D enclosed by C. C must be a positively oriented (counterclockwise), piecewise smooth, simple closed curve. P and Q must have continuous partial derivatives on an open region containing D.
Variables
| Symbol | Meaning |
|---|---|
| C | Positively oriented closed curve (boundary of D) |
| D | Region enclosed by C |
| P(x, y) | Component of the vector field in the x-direction |
| Q(x, y) | Component of the vector field in the y-direction |
| ∂Q/∂x − ∂P/∂y | 2D curl of the vector field F = (P, Q) |
Area formula using Green's theorem:
Setting P = 0, Q = x gives: A = ∮ x dy
Setting P = −y, Q = 0 gives: A = ∮ (−y) dx = ∮ y dx taken as negative
Symmetric form: A = ½ ∮C (x dy − y dx)
Example 1 — Area of an Ellipse
Use Green's theorem to find the area enclosed by the ellipse x = a cos(t), y = b sin(t), t from 0 to 2π.
A = ½ ∮ (x dy − y dx)
dx = −a sin(t) dt, dy = b cos(t) dt
x dy − y dx = a cos(t) × b cos(t) dt − b sin(t) × (−a sin(t)) dt
= ab cos²(t) dt + ab sin²(t) dt = ab dt
A = ½ ∫02π ab dt = ½ × ab × 2π = πab — the standard area formula for an ellipse!
Example 2 — Verifying Green's Theorem
Verify Green's theorem for P = x², Q = xy over the unit square [0,1] × [0,1].
Right side: &iint; (∂(xy)/∂x − ∂(x²)/∂y) dA = &iint; (y − 0) dA = ∫01 ∫01 y dy dx = ∫01 ½ dx = ½
Left side: compute line integral over 4 edges of the square
Bottom (y=0, dx): ∫01 x² dx + 0 = 1/3
Right (x=1, dy): 0 + ∫01 1×y dy = 1/2; Top (y=1, dx): ∫10 x² dx = −1/3; Left (x=0): 0
Total = 1/3 + 1/2 − 1/3 + 0 = 1/2 ✓ Both sides equal ½
When to Use It
Use Green's theorem when:
- The line integral around a closed curve is difficult but the double integral is easier (or vice versa)
- Calculating areas of irregular regions using the boundary-area formula
- Proving conservation laws in 2D vector fields (if curl = 0, the field is conservative)
- Solving problems in fluid mechanics and electromagnetism in 2D