Use Stoke's theorem to evaluate β«β«ScurlFβ
dS where F(x,y,z)=β17yzi+17xzj+17(x2+y^2)zk?
and S is the part of the paraboloid z=x^2+y^2 that lies inside the cylinder x^2+y^2=1, oriented upward
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and S is the part of the paraboloid z=x^2+y^2 that lies inside the cylinder x^2+y^2=1, oriented upward
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These surfaces meet in the circle πͺ, xΒ²+yΒ²=1, z=1
By Stokes, β¬[π] curl(π)β’π π = β«[πͺ] πβ’π π where normal is upwards for π and πͺ is anti-clockwise
β«[πͺ] πβ’π π = β«[πͺ] 17( βyz, xz, (xΒ²+yΒ²)z )β’( dx. dy. 0 ) = 17 β«[πͺ] (βyzdx+xzdy) = 17 β«[πͺ] (βydx+xdy)
By Greenβs Theorem this is 17 β¬ (1β(β1))dxdy over the interior of πͺ = 34 (area inside πͺ) = 34Ο