Verifying Stokes' for F = <-y, x, z> over Hemisphere

Calculus 3 · Axiom Academy

EXAMPLE Verifying Stokes' Theorem Calculate circulation directly and via curl to verify Stokes' Theorem for a hemisphere Verify Stokes' Theorem, , for the vector field below by computing both the line integral around the boundary and the surface integral of the curl, and checking that they agree. Surface S : the upper hemisphere Boundary C : the unit circle , traversed counterclockwise as seen from above Left — the boundary circle C viewed from above: traced counterclockwise as increases, matching the orientation Stokes' Theorem requires for an upward-pointing normal. Right — a vertical slice through the hemisphere: is the angle from the z -axis, and the outward normal always points straight out along the radius, away from the center. Excellent work! You've successfully verified Stokes' Theorem. Here's what we learned: Two Paths, Same Result: The line integral around the boundary ( ) equals the surface integral of the curl ( ), confirming Stokes' Theorem. Boundary Parameterization: For a circle in the xy -plane, use for . Computing Curl: For , the curl is — constant rotation about the z -axis. Surface Parameterization: The hemisphere uses spherical coordinates, . Normal Vector: The cross product gives the outward normal, essential for matching the boundary's orientation. Practical Advantage: Stokes' Theorem lets us choose the easier calculation — sometimes the line integral is simpler, sometimes the surface integral is!

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