Green’s Theorem Calculator
Calculate Circulation and Flux around a closed region.
1. Vector Field F(x, y)
2. Region D (Rectangle)
The tool calculates the double integral over this area.
Total Circulation
Field & Region Visualization
Understanding Green’s Theorem
Green’s Theorem is a stunning connection between the microscopic behavior of a field inside a region and the macroscopic flow around its edge. It tells us that adding up all the tiny swirls (curl) inside a shape is exactly equal to the total swirl along the boundary.
1. Circulation Form (Curl)
Used to find the work done by a force moving around a closed loop.
∮C F · dr = ∬D (∂Q/∂x – ∂P/∂y) dA
The term in the double integral is the 2D Curl.
2. Flux Form (Divergence)
Used to find the total fluid flowing outward across a closed boundary.
∮C F · n ds = ∬D (∂P/∂x + ∂Q/∂y) dA
The term in the double integral is the Divergence.
The “Cancellation” Analogy
Imagine the region D is filled with tiny spinning paddle wheels. Where two wheels touch, one spins clockwise against the other spinning counter-clockwise. Their edges cancel out! The only part that doesn’t cancel is the very outer edge of the region.
This is why summing the internal curls (Double Integral) equals the boundary circulation (Line Integral).
Frequently Asked Questions (FAQ)
Q: Does orientation matter?
A: Yes! Green’s Theorem assumes the curve C is oriented Counter-Clockwise (Positive Orientation). If you go clockwise, the result is negative.
Q: What if the region has a hole?
A: Green’s Theorem still works, but the boundary C includes both the outer edge (counter-clockwise) and the inner hole edge (clockwise).
Q: Why calculate the double integral instead?
A: Often, the line integral is hard because the boundary has many jagged pieces. The double integral over the area might be a simple constant or polynomial, making it much faster to solve.