Green's Theorem is a powerful result in vector calculus that connects motion around a closed curve to behavior throughout the region inside the curve. Instead of adding up a vector field along a boundary directly, you can often compute an area integral over the enclosed region. This matters because it turns difficult line integrals into more manageable double integrals.
It also gives a clear geometric meaning to circulation in a plane.
Key Facts
- Green's Theorem circulation form: ∮C F · dr = ∮C P dx + Q dy = ∬R (∂Q/∂x - ∂P/∂y) dA
- For F = <P, Q>, the scalar curl in the plane is curl F = ∂Q/∂x - ∂P/∂y
- Positive orientation means the curve C is traveled counterclockwise, with the region R on the left
- Green's Theorem applies when C is a simple closed curve and P, Q have continuous first partial derivatives on a region containing R
- If ∂Q/∂x - ∂P/∂y = 0 throughout R, then the circulation around C is 0
- Area can be found using Green's Theorem: Area(R) = ∮C x dy = -∮C y dx = 1/2 ∮C (x dy - y dx)
Vocabulary
- Line integral
- A line integral adds the values of a function or vector field along a curve.
- Closed curve
- A closed curve is a path that starts and ends at the same point.
- Positive orientation
- Positive orientation means traveling counterclockwise around a region so the region stays on your left.
- Circulation
- Circulation measures how much a vector field tends to flow around a closed curve.
- Scalar curl
- Scalar curl in the plane is ∂Q/∂x - ∂P/∂y and measures local rotation of a two-dimensional vector field.
Common Mistakes to Avoid
- Using the wrong orientation, because Green's Theorem in circulation form assumes positive counterclockwise orientation and a clockwise path changes the sign.
- Swapping the partial derivatives, because the correct scalar curl is ∂Q/∂x - ∂P/∂y, not ∂P/∂y - ∂Q/∂x.
- Applying the theorem to a curve that is not closed, because Green's Theorem relates a closed boundary integral to an area integral over the enclosed region.
- Forgetting to check smoothness and holes, because discontinuities inside the region or multiply connected regions may require splitting the region or accounting for extra boundary curves.
Practice Questions
- 1 Use Green's Theorem to compute ∮C (-y dx + x dy), where C is the circle x^2 + y^2 = 9 oriented counterclockwise.
- 2 Use Green's Theorem to compute ∮C (x^2 dx + xy dy), where C is the rectangle 0 ≤ x ≤ 2, 0 ≤ y ≤ 3 oriented counterclockwise.
- 3 A vector field has positive scalar curl everywhere inside a simple closed curve. Explain what Green's Theorem predicts about the sign of the counterclockwise circulation around the curve.