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MathCalculus

Consider the Following Region R and Vector Field F: Verify Both Forms of Green's Theorem

Verify Green's theorem on the unit disk for F=(x-y,x) by evaluating both the circulation and outward-flux line and area integrals.

Question

For F(x,y)=(x−y,x)\mathbf F(x,y)=(x-y,x) and the disk R:x2+y2≤1R:x^2+y^2\le1, with its boundary CC parameterized by r(t)=(cos⁡t,sin⁡t)\mathbf r(t)=(\cos t,\sin t) for 0≤t≤2π0\le t\le2\pi, verify both the circulation and the outward-flux forms of Green's theorem. Evaluate both sides of each identity separately.

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Evidence boundary

The source fixes the vector field, unit-circle boundary, and direction through its parameterization. The four integral values above are independently calculated from those data; they are not copied from an answer key. The result assumes the counterclockwise orientation t=0→2πt=0\to2\pi.