Exercise · ZW-GREENS-THEOREM-005 · ★☆☆ Basic
Compute \(\oint_C y^3\,dx+x^3\,dy\) over the unit circle, counterclockwise.
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- B. \(0\)
Explanation
\(Q_x-P_y=3x^2-3y^2\), and by symmetry of the disk \(\iint_D x^2\,dA=\iint_D y^2\,dA\); the integral is \(0\).
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
- ZW-GREENS-THEOREM-001 Let C be the circle x^2+y^2=4 traversed counterclockwise. Compute oint_C -y dx+x dy.
- ZW-GREENS-THEOREM-002 Let C be the boundary of the rectangle [0,3]×[0,2], traversed counterclockwise. Compute oint_C y dx.
- ZW-GREENS-THEOREM-003 Compute oint_C xy dx+x^2 dy, where C is the counterclockwise boundary of the square [0,1]×[0,1].
- ZW-GREENS-THEOREM-004 Compute oint_C x^2 dx+xy dy, where C is the counterclockwise boundary of the triangle with vertices…
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