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Multivariable and Vector Calculus · Vector fields, line integrals, and Green's theorem

Check outward flux across a parabolic cap

Problem

Let \[ D=\{(x,y):0\le x\le1,\ 0\le y\le x(1-x)\} \] and let $C=\partial D$ have positive counterclockwise orientation. For \[ \mathbf F(x,y)=(P,Q)=(x^2y,xy^2), \] compute the total outward flux across $C$ using the flux form of Green's theorem. Then compute the same flux directly from \[ \oint_C P\,dy-Q\,dx \] by treating the lower segment and the parabolic arc separately. State the correct direction of traversal on each piece, reconcile the two exact results, and explain why substituting the circulation integrand $P\,dx+Q\,dy$ would answer a different question.

Hint

The divergence is $P_x+Q_y=4xy$.

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