Verifying Green's Theorem for F = <y², x> over Circle

Calculus 3 · Axiom Academy

EXAMPLE Verifying Green's Theorem Calculate both the line integral and double integral to verify Green's Theorem for a vector field over a circular region. Verify Green's Theorem for the vector field over the region enclosed by C , the unit circle x^2 + y^2 = 1 traversed counterclockwise — that is, confirm that the line integral around C equals the double integral of the curl over the disk D : Excellent work! You've successfully verified Green's Theorem. Here's what we learned: Green's Theorem connects line and double integrals: it relates the circulation around a closed curve to the curl integrated over the region it encloses. Parametrization is crucial: we used to parametrize the circle, making the line integral tractable. Computing the curl: for , the curl is . In this case, 1-2y becomes in polar coordinates. Polar coordinates simplify disk integrals: converting to polar with the Jacobian r made the integration straightforward. Verification confirms the theorem: both methods yielded , confirming Green's Theorem holds for this vector field and region! This powerful theorem lets us convert difficult line integrals into easier double integrals (or vice versa) — a cornerstone result in vector calculus.

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