Math101 · Independent Ontario learning library
Math101Green's Theorem
A rigorous, example-driven guide to green's theorem, including hypotheses, method choice, verification, and practice.
The central idea
Let $C$ be a positively oriented, piecewise smooth, simple closed curve bounding a planar region $D$, and let $P,Q$ have continuous partial derivatives on an open set containing $D$. Then $\oint_C Pdx+Qdy=\iint_D(Q_x-P_y)dA$. The flux form is $\oint_C Pdy-Qdx=\iint_D(P_x+Q_y)dA$.
Definitions, hypotheses, and notation
Multiply connected regions require every boundary component with induced orientation: the outer boundary is counterclockwise and hole boundaries clockwise, keeping the region on the left. If a field is singular in a hole excluded from $D$, the theorem may still apply on the remaining region, but not across the singular point.
Area can be computed from line integrals such as $A=\frac12\oint_C xdy-ydx$. This follows by choosing a field with scalar curl one. Different choices yield equivalent area formulas, and orientation controls whether the signed result is positive.
Conceptual meaning
Green's theorem converts circulation or flux around a boundary into accumulated local rotation or divergence across the interior. Positive orientation is counterclockwise: as the boundary is traversed, the region stays on the left.
A dependable method and decision rule
- Verify $C$ is closed, simple or properly decomposed, and positively oriented.
- Choose circulation or flux form from the differential expression.
- Compute the scalar curl or planar divergence.
- Describe and integrate over $D$ in suitable coordinates.
- Reverse the sign if the original curve is clockwise.
