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Green's theorem in the plane.

Theorem 8.20 (Flux-Divergence form)   Let be a vector field over a region in the plane. Denote by a simple loop in , and by the region enclosed by this loop.

The outward flux of across is equal to the double integral of over :

Example 8.21   We verify theorem  8.20 for over the region bounded by the circle whose parametrization is .
• Components of the field:

• Differentials:

• Partial derivatives:

• Left side of Eq.  8.20:

• Right side of Eq.  8.20:

Theorem 8.22 (Circulation-Curl form)   Let be a vector field over a region in the plane. Denote by a simple loop in , and by the region enclosed by this loop.

The counterclockwise circulation of along is equal to the double integral of over :

Example 8.23   We verify theorem  8.22 for over the region bounded by the circle whose parametrization is .
• Components of the field:

• Differentials:

• Partial derivatives:

• Left side of Eq.  8.22:

• Right side of Eq.  8.22:

Example 8.24   Compute the integral , where is the border of the annulus defined by .

1.
Compute the line integral: The border of the annulus is the union of two circles:
• whose equation is x2+y2=9; on , we turn counterclockwise, so a suitable parametrization is , where
• whose equation is x2+y2=1; on , we turn clockwise, so a suitable parametrization is , where
Now we compute the two line integrals:

Thus we have:

I=I1+I2= 0.

2.
With Green's theorem:

We have: M= xy+y2 and N=x2-y. Then:

We use theorem  8.22:

Next: Surface integrals and Surface Up: Integration and Vector Fields. Previous: Flow, circulation and flux.
Noah Dana-Picard
2001-05-30