Circulation

**Apprentice123** · June 3rd 2009, 04:27 PM

Find circulation
$F(x,y) = (x+2xy)i + (y+x^2)j$ around the

$\alpha : |x| + |y| = 1$

**Random Variable** · June 3rd 2009, 04:47 PM

Use Green's Theorem which says the circulation of a vector field around a simpled closed curve in the xy-plane is equal to the line integral of the vector field evaluated around the curve. In ths problem the curve is diamond-shaped. So you have to parametrize each side and evaluate four separate line integrals.

BTW, is the answer to that flux problem in your textbook?

**Random Variable** · June 3rd 2009, 06:28 PM

Starting at (1,0) and and going counterlcockwise

$\int \int_{D} (\nabla \times F) \cdot \hat{k} \ dA$ = $\int_{C} F \cdot dr = \int_{0}^{1} \Big( F(1-t,t) \cdot (-1,1) \Big) dt + \int_{0}^{1} \Big(F(-t,1-t) \cdot (-1,-1) \Big) dt$ $+ \int_{0}^{1} \Big(F(t-1, -t) \cdot (1,-1) \Big) dt + \int_{0}^{1} \Big( F(t, t-1) \cdot (1,1) \Big) dt$

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