Exercise · ZW-GREENS-THEOREM-003 · ★☆☆ Basic
Compute \(\oint_C xy\,dx+x^2\,dy\), where \(C\) is the counterclockwise boundary of the square \([0,1]\times[0,1]\).
Choose one option.
Show answer
- C. \(\tfrac12\)
Explanation
\(Q_x-P_y=2x-x=x\) and \(\int_0^1\!\int_0^1 x\,dy\,dx=\tfrac12\).
Review the concept: Green's Theorem
Green's theorem equates the circulation of a field \(\mathbf F=\langle P,Q\rangle\) along a simple closed curve \(C\), counterclockwise, with a double integral over its region \(D\), with respect to area \(A\).
\[\oint_C P\,dx+Q\,dy=\iint_D\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dA\]
More practice problems: green's theorem
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