Complete the statement of Green’s theorem.
Green’s theorem converts the circulation around a closed curve into the double integral over the enclosed region of ______ — a local rotation density. The curve must be traversed ______, and the theorem breaks if the field has a ______ inside the region.
Word bank: Qₓ − Pᵧ (the 2D curl) · counterclockwise · singularity · Pₓ + Qᵧ (the divergence) · clockwise
Hints
- The integrand is the 2D curl, the derivative comes first, with a minus on the term.
- Points where the field blows up (like at the origin) puncture the region and break the theorem’s hypotheses.
Show the answer
Green’s theorem converts the circulation around a closed curve into the double integral over the enclosed region of Qₓ − Pᵧ (the 2D curl) — a local rotation density. The curve must be traversed counterclockwise, and the theorem breaks if the field has a singularity inside the region.
Green’s theorem equates boundary circulation with the total of the 2D curl inside. The hypotheses have teeth: one singular point inside (think of a vortex at the origin) and the two sides genuinely disagree, the basis for winding numbers in complex analysis.
Practise Green's Theorem
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