Verify Stokes' Theorem for the given vector field and surface, oriented with an upwardpointing normal.

\(\mathbf{F}=\left\langle y, x, x^{2}+y^{2}ightangle\), the upper hemisphere \(x^{2}+y^{2}+z^{2}=1, z \geq 0\)

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THEOREM 1 Stokes' Theorem Let S be a surface as described earlier, and let F be a vector field whose components have continuous partial derivatives on an open region containing S. as F.dr = JG Ils The integral on the left is defined relative to the boundary orientation of S. If S is a closed surface, then curl(F). ds curl(F) dS=0 1