Exercise · ZW-GREENS-THEOREM-002 · ★☆☆ Basic
Let \(C\) be the boundary of the rectangle \([0,3]\times[0,2]\), traversed counterclockwise. Compute \(\oint_C y\,dx\).
Choose one option.
Show answer
- A. \(-6\)
Explanation
With \(P=y\) and \(Q=0\), \(Q_x-P_y=-1\); the integral is \(-1\) times the area \(6\), that is \(-6\).
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 \(A\).
\[\oint_C P\,dx+Q\,dy=\iint_D\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dA\]
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