Green's Theorem: Circulation, Flux, Proof, and Area

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State Green's theorem in circulation form.

For a positively oriented simple closed curve $C$ bounding $D$, with $L$ and $M$ having continuous first partial derivatives, $\oint_C(L\,dx+M\,dy)=\iint_D\left(\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y}\right)\,dA$.

What is positive orientation?

It is counterclockwise traversal of the boundary. As the curve is followed, the enclosed region lies on the left.

What is the scalar curl of $(L,M)$?

The scalar curl is $\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y}$. It measures local rotation in the plane.

What is a simple closed curve?

A curve that returns to its starting point without crossing itself; it encloses a planar region.

Why check the orientation before using Green's theorem?

The theorem's standard form assumes counterclockwise orientation. Reversing the direction reverses the sign of the boundary line integral.

Quiz questions

1. For a planar vector field $\mathbf F=(P,Q)$, what integrand appears in the region integral of Green's flux form?Answer: $\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}$
2. Which rotated field can be used with Green's circulation form to derive the flux form for $\mathbf F=(P,Q)$?Answer: $(Q,-P)$
3. When the planar field $(L,M)$ is embedded as $(L,M,0)$ on the $xy$-plane, which component of its three-dimensional curl gives Green's circulation integrand?Answer: The vertical component, $\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y}$
4. When a region is divided into simpler pieces to apply Green's theorem, what happens to line integrals along shared internal boundaries?Answer: They cancel because adjacent pieces traverse each shared boundary in opposite directions.
5. For a region with holes, how is its boundary oriented when applying Green's theorem to the region itself?Answer: The outer boundary is counterclockwise, and each hole boundary is clockwise.